GG¨odel Diffeomorphisms
Matthew Foreman ∗† September 22, 2020
Abstract
A basic problem in smooth dynamics is determining if a system can bedistinguished from its inverse, i.e., whether a smooth diffeomorphism T is iso-morphic to T − . We show that this problem is sufficiently general that askingit for particular choices of T is equivalent to the validity of well-known numbertheoretic conjectures including the Riemann Hypothesis and Goldbach’s con-jecture. Further one can produce computable diffeomorphisms T such that thequestion of whether T is isomorphic to T − is independent of ZFC. Contents -sets and G¨odel numberings . . . . . . . . . . . . . . . . . . . . . 71.4 Effectively computable diffeomorphisms . . . . . . . . . . . . . . . . 81.5 Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . 10 F O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Properties of the words and actions. . . . . . . . . . . . . . . . . . . 212.6 Building the words, equivalence relations and actions . . . . . . . . . 24 ∗ University of California, Irvine. † Foreman’s research was supported by the National Science foundation grant DMS-1700143 a r X i v : . [ m a t h . D S ] S e p Circular Systems and Diffeomorphisms of the Torus 32 T n . . . . . . . . . . . . . . . . . . . . . . . . 533.3.3 The effective computation of S n . . . . . . . . . . . . . . . . 563.4 Completing the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 A Numerical Parameters 60
A.1 The Numerical Requirements Collected. . . . . . . . . . . . . . . . . 60A.2 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
B Logical Background 70
B.1 Logical Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70B.1.1 The language L PA . . . . . . . . . . . . . . . . . . . . . . . . 70B.1.2 Bounded quantifiers . . . . . . . . . . . . . . . . . . . . . . . 72B.1.3 Formula complexity . . . . . . . . . . . . . . . . . . . . . . . 73B.1.4 Π -formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74B.1.5 Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75B.2 Computability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 76B.2.1 Primitive recursion . . . . . . . . . . . . . . . . . . . . . . . . 76B.2.2 Computable real functions . . . . . . . . . . . . . . . . . . . . 77B.2.3 Modulus of continuity and approximation . . . . . . . . . . . 78 C Ergodic Theory Background 80
C.1 Why Z ? Why T ? Why C ∞ ? . . . . . . . . . . . . . . . . . . . . . . 80C.2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81C.3 Symbolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82C.4 Odometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84C.5 Factors, joinings, and conjugacies . . . . . . . . . . . . . . . . . . . . 86 D Diffeomorphisms 87
D.1 Diffeomorphisms of the torus . . . . . . . . . . . . . . . . . . . . . . 88D.2 Smooth permutations . . . . . . . . . . . . . . . . . . . . . . . . . . 89
When is forward time isomorphic to backward time for a given dynamical system?When the acting group is Z , this asks when a transformation T is isomorphic to2ts inverse. It was not until 1951, that Anzai [2] refuted a conjecture of Halmosand von Neumann by exhibiting the first example of a transformation where T isnot measure theoretically isomorphic to its inverse. In fact the general problem isso complex that it cannot be be resolved using an arbitrary countable amount ofinformation: in [11], it was shown that the collection of ergodic Lebesgue measurepreserving diffeomorphisms of the 2-torus isomorphic to their inverse is completeanalytic and hence not Borel.In this paper we show that for a broad class of problems there is a one-to-onecomputable method of associating a Lebesgue measure preserving diffeomorphism T P of the two-torus to each problem P in this class so that: • P is trueif and only if • T P is measure theoretically isomorphic to T − P .The class of problems is large enough to include the Riemann Hypothesis , Gold-bach’s Conjecture and statements such as “
Zermelo-Frankel Set Theory (
ZFC ) isconsistent. ” In consequence, each of these problems is equivalent to the question ofwhether T ∼ = T − for the diffeomorphism T of 2-torus canonically associated to thatproblem.Restating this, there is an ergodic diffeomorphism of the two-torus T RH suchthat T RH ∼ = T RH − if and only if the Riemann Hypothesis holds, and a different,non-isomorphic ergodic diffeomorphism T GC such that T GC ∼ = T GC − if and only ifGoldbach’s conjecture holds, and so forth.G¨odel’s Second Incompleteness Theorem states that for any recursively axioma-tizable theory Σ that is sufficiently strong to prove basic arithmetic facts, if Σ provesthe statement “Σ is consistent ”, then Σ is in fact inconsistent . The statement “Σ isconsistent” can be formalized in the manner of the problems we consider. Considerthe most standard axiomatization for mathematics: Zermelo-Frankel Set Theorywith the Axiom of Choice and the formalization of its consistency, the statementCon(ZFC).If T ZFC is the diffeomorphism associated with Con(ZFC) then (assuming theconsistency of conventional mathematics) the question of whether T ZFC ∼ = T ZFC − is independent of Zermelo-Frankel Set Theory—that is, it cannot be settled withthe usual assumptions of mathematics.One can compare this with more standard independence results, the most promi-nent being the Continuum Hypothesis. Those independence results inherently in-volve comparisons between and properties of uncountable objects. The results inthis paper are about the relationships between finite computable objects. See for example Math Review MR0047742 where Halmos states “By constructing an exampleof the type described in the title the author solves (negatively) a problem proposed by the reviewerand von Neumann [Ann. of Math. (2) 63, 332–350 (1962); MR0006617].”
3e now give precise statements of the main theorem and its corollaries. Themachinery for proving these results combines ergodic theory and descriptive set the-ory with logical and meta-mathematical techniques originally developed by G¨odel.While the statements use only standard terminology, it is combined from severalfields. We have included several appendices in an attempt to convey this back-ground to non-experts.There are several standard references for connections between non-computablesets and analysis and PDE’s. We note one in particular with results of MarianPour-El and Ian Richards that give an example of a wave equation with computableinitial data but no computable solution [21].
As an informal guide to reading the theorem, we say a couple of words. More formaldefinitions appear in later sections. • A function F being computable means that there is a computer program thaton input N outputs F ( N ). • The diffeomorphisms in the paper are taken to be C ∞ and Lebesgue measurepreserving. A diffeomorphism T is computable if there is a computer programthat when serially fed the decimal expansions of a pair ( x, y ) ∈ T outputs thedecimal expansions of T ( x, y ) and for each n there is a computable functioncomputing the decimal expansion of the modulus of continuity of the n -thdifferential. Since computable functions have codes, computable diffeomor-phisms also can be coded by natural numbers. • By isomorphism, it is meant measure isomorphism . Measure preserving trans-formations S : X → X and T : Y → Y are measure theoretically isomorphicif there is a measure isomorphism ϕ : X → Y such that S ◦ ϕ = T ◦ S up to a sets of measure zero. • In Appendix C we discuss questions such as
Why Z ? Why T ? Why C ∞ ? .We use the notation Diff ∞ ( T , λ ) for the collection of C ∞ measure-preservingdiffeomorphisms of T . • Π statements are those number-theoretic statements that start with a blockof universal quantifiers and are followed by Boolean combinations of equalitiesand inequalities of polynomials with natural number coefficients. Recent work of Banerjee and Kunde in [3] allow Theorem 1 to be extended to real analyticfunctions by improving the realization results in [13]. We fix G¨odel numberings: computable ways of enumerating Π statements (cid:104) ϕ n : n ∈ N (cid:105) and computer programs (cid:104) C m : m ∈ N (cid:105) . The code of ϕ n is n , the code of C m is m . • Older literature uses the word recursive and more recent literature uses theword computable as a synonym. We use the latter in this paper. Indeed, sincenone of the phenomenon discussed here involve recursive behavior that is notprimitive recursive we use effective , and computable as synonyms for primitiverecursive.
Here is the statement of the main theorem.
Theorem 1. (Main Theorem)
There is a computable function F : { Codes for Π -sentences } → { Codes for computable diffeomorphisms of T } such that:1. N is the code for a true statement if and only if F ( N ) is the code for T , where T is measure theoretically isomorphic to T − ;2. For M (cid:54) = N , F ( M ) is not isomorphic to F ( N ) .The diffeomorphisms in the range of F are Lebesgue measure preserving and ergodic. We now explicitly draw corollaries.
Corollary 2.
There is an ergodic diffeomorphism of the two-torus T RH such that T RH ∼ = T − RH if and only if the Riemann Hypothesis holds. Similarly:
Corollary 3.
There is an ergodic diffeomorphism of the two-torus T GC such that T GC ∼ = T − GC if and only if Goldbach’s Conjecture holds. There are at least two reasons that this theorem is not trivial. The first is thatthe function F is computable, hence the association of the diffeomorphism to theΠ statement is canonical. Secondly the function is one-to-one; T RH encodes theRiemann hypothesis and T GC encodes Goldbach’s conjecture and T RH (cid:54)∼ = T GC . Corollary 4.
Assume that
ZFC is consistent. Then there is a computable ergodicdiffeomorphism T of the torus such that T is measure theoretically isomorphic to T − , but this is unprovable in Zermelo-Frankel set theory together with the Axiomof Choice.
5e note again that there is nothing particularly distinctive about Zermelo-Frankel set theory with the Axiom of Choice. We choose it for the corollary becauseit forms the usual axiom system for mathematics. Thus Corollary 4 states an inde-pendence result in a classical form. Similar results can be drawn for theories of theform “
ZFC + there is a large cardinal” or simply ZF without the Axiom of Choice.Finally, these results can be modified quite easily to produce diffeomorphismsof (e.g.) the unit disc with the analogous properties. Moreover techniques fromthe thesis of Banerjee ([4]) and Banerjee-Kunde ([3]) can be used to improve thereduction F so that the range consists of real analytic maps of the 2-torus.We finish this section by thanking Tim Carlson for asking whether Theorem 1can be extended to lightface Σ statements, which it can in a straightforward way.This increases the collection of statements encoded into diffeomorphisms to includevirtually all standard mathematical statements. Primitive recursion
Informally, primitive recursive functions are those that canbe computed by a program that uses only for statements and not while statements.This means that the computational time can be bounded constructively using it-erated exponential maps. In the statements of the results we discuss “computablefunctions” but in fact all of the functions constructed are primitive recursive. In par-ticular the functions and computable diffeomorphisms asserted to exist in Theorem1 are primitive recursive.
Hilbert’s 10th problem asks for a general algorithm for deciding whether Diophantineequations have integer solutions. The existence of such an algorithm was shown tobe impossible by a succession of results of Davis, Putnam and Robinson culminatinga complete solution by Matijaseviˇc in 1970 ([18, 6]).Their solution can be recast as a statement very similar to Theorem 1:There is a computable function F : { Codes for Π -sentences } → { Diophantine Polynomials } such that N is the code for a true statement if and only if F ( N ) has nointeger solutions.Thus their theorem reduces general questions about the truth of Π statements toquestions about zeros of polynomials. Theorem 1 states that there is an effectivereduction of the true Π statements to C ∞ transformations isomorphic to theirinverse. 6 .3 Π -sets and G¨odel numberings While the interesting corollaries of Theorem 1 are about the Riemann Hypothesis,other number theoretic statements, and independence results for dynamical systems,it is actually a theorem about subsets of N . In order to prove it, one has to pro-vide a way of translating between the interesting mathematical objects as they areusually constructed and the natural numbers that encode them. This is done bymeans of G¨odel numberings , natural numbers which code the structure of familiarmathematical objects.The arithmetization of syntax via
G¨odel Numbers is one of the main insightsin the proofs of the Incompleteness Theorems. It is used to state “Σ is consistent”(where Σ is an enumerable set of axioms) as a Π statement. G¨odel numberingsoriginally appear in [16], but are covered in any standard logic text such as [8].The idea behind G¨odel numberings is very simple: let (cid:104) p n : n ∈ N (cid:105) be anenumeration of the prime numbers. Associate a positive integer to each symbol:“ x ” might be 1, “0” might be 2, “ ∀ ” might be 3 and so on. Then a sequence ofsymbols of length k can be coded as c = 2 n · n · n · · · p n k k . Example 5.
Suppose we use the following coding scheme:
Symbol x 0 ∀ ∗ = ( )Integer ∀ x ( x ∗ is c = 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ . Clearly the sentence can be uniquely recovered from its code. With more work,one can also use natural numbers to effectively code computer programs and theircomputations, sequences of formulas that constitute a proof and many other objects.The methods use the Chinese Remainder Theorem.We now turn to Π sentences. Definition 6.
A sentence ϕ in the language L PA = { + , ∗ , , , < } is Π if it canbe written in the form ( ∀ x )( ∀ x ) . . . ( ∀ x n ) ψ , where ψ is a Boolean combination ofequalities and inequalities of polynomials in the variables x , . . . x n and the constants , . (We do not allow unquantified—i.e., free —variables to appear in ϕ .) It is not difficult to show that { n : n is the G¨odel number of a Π sentence in a finite language } is a computable set.It is however, non-trivial to show that some statements such as the RiemannHypothesis and the consistency of ZFC are provably equivalent to Π -statements.7he Riemann Hypothesis was shown to be Π by Davis, Matijaseviˇc and Robin-son ([6]) and a particularly elegant version of such a statement is due to Lagarias([17]). Appendix B.1.4 exhibits Π -statements that are equivalent to the RiemannHypothesis (using [17]) and Goldbach’s Conjecture. Truth:
We say a sentence ϕ in the language L P A is true if it holds in the structure( N , + , ∗ , , , < ). Since T is compact, a C ∞ -diffeomorphism T is uniformly continuous, as are itsdifferentials. Thus, it makes sense to view their moduli of continuity as functions d : N → N which say, informally, that if one wishes to specify the map ( x, y ) (cid:55)→ T ( x, y )to within 2 − n , then the original point ( x, y ) must be specified to within a toleranceof 2 − d ( n ) . With better and better information about ( x, y ), one can produce betterand better information about T ( x, y ). This intuitive notion is formalized by thedefinitions given below, and in more detail in Appendix B.2.2. We note in passingthat the moduli of continuity and approximations are not uniquely defined.
Definition 7 (Effective Uniform Continuity) . We say that a map T : T → T is effectively uniformly continuous if and only if the following two computable functionsexist: • A computable Modulus of Continuity:
A computable function d : N → N which, given a target accuracy (cid:15) finds the δ within which the source must beknown to approximate the function within (cid:15) .More concretely, suppose T : [0 , × [0 , → [0 , × [0 , . View elements in [0 , as their binary expansions. Then the first d ( n ) digits of each of ( x, y ) determine the first n digits of the two entries of T ( x, y ) . • A Computable Approximation:
A computable function f : ( { , }×{ , } ) < N → ( { , } × { , } ) < N , which, given the first d ( n ) digits of the binary expansionof ( x, y ) —or, equivalently, the dyadic rational numbers ( k x · − d ( n ) , k y · − d ( n ) ) for ≤ k x , k y ≤ d ( n ) closest to ( x, y ) — outputs the first n digits of the binaryexpansion of the coordinates of T ( x, y ) . The diffeomorphisms T we build are C ∞ and map from T to T . Because weare working on T we can view T as a map from R to R . The k th differential isdetermined by the collection of k th partial derivatives { ∂ k ∂ i x∂ k − i y : 0 ≤ i ≤ k } of T with respect to the standard coordinate system for R . For k < ∞ , T is effectively Since diffeomorphisms are Lipshitz, we could have worked with computable Lipshitz constantsrather then computable moduli of continuity. The methods give the same collections of computablediffeomorphisms. k provided that for each n < k there are computable d ( n, − ) and f ( n, − ) that givethe moduli of continuity and approximations to the partial n th derivatives. Being C ∞ requires that the d ( n, − ) and f ( n, − ) exist and are uniformly computable; thatis that there is a single algorithm that on every input n ∈ N computes d ( n, − ) and f ( n, − ).For clarity, in these definitions we discussed functions with domain and range T . There is no difficulty generalizing effective uniform continuity to effectivelypresented metric spaces. The notion of a computable C k diffeomorphism also easilygeneralized to smooth manifolds M and their diffeomorphisms, using atlases.We note that computable diffeomorphisms are uniquely determined by the proce-dures for computing d and f and hence they too may be coded using G¨odel numbers.The elements of the range of the function F in Theorem 1 code diffeomorphisms inthis manner. Inverses of recursive diffeomorphisms
It is not true that the inverse of aprimitive recursive function f : N → N is primitive recursive. However for primitiverecursive diffeomorphisms of compact manifolds it is. Suppose that M is a smoothcompact manifold and T is a C ∞ -diffeomorphism. Then T is a diffeomorphism andhence has uniformly Lipschitz differentials of all orders. Since T is invertible and M is compact, T − also has uniformly Lipschitz differentials of all orders. Moreover theLipschitz constants for T − are “one over” the Lipschitz constants for T . It followsin a straightforward way that the inverse of a primitive recursive diffeomorphism on M is a primitive recursive diffeomorphism. The key idea for proving Theorem 1 is that of a reduction . Definition 8.
Suppose that A ⊆ X and B ⊆ Y and f : X → Y . Then f reduces A to B if x ∈ A iff f ( x ) ∈ B. The idea behind a reduction is that to determine whether a point x belongs to A one looks at f ( x ) and asks whether it belongs to B : f reduces the question “ x ∈ A ”to “ f ( x ) ∈ B .For this to be interesting the function f must be relatively simple. In manycases the spaces X and Y are Polish spaces and f is taken to be a Borel map. Inthis paper X = Y = N and F is primitive recursive.In [11] the function f has domain the space of trees (equivalently, acyclic count-able graphs) and has range the space of measure preserving diffeomorphisms ofthe two-torus. It reduces the collection of ill-founded trees (those with an infinitebranch or, respectively, acyclic graphs with a non-trivial end) to diffeomorphismsisomorphic to their inverse. 9he function f is a Borel map. The point there is that if { T : T ∼ = T − } wereBorel then its inverse by the Borel function f would also have to be Borel. But theset of ill-founded trees is known not to be Borel. Hence the isomorphism relation ofdiffeomorphisms is not Borel.In the current context the function F in Theorem 1 maps from a computablesubset of N (the collection of G¨odel numbers of Π statements) to N . It takes valuesin the collection of codes for diffeomorphisms of the two-torus.Theorem 1 can be restated as saying that F is a primitive recursive reductionof the collection A of G¨odel numbers for true Π statements to the collection B of codes for computable measure preserving diffeomorphisms of the torus that areisomorphic to their inverses. For N (cid:54) = M the transformation F ( N ) is not isomorphicto F ( M ).Thus Theorem 1 can be restated as saying that the collection of true Π state-ments is computably reducible to the collection of measure preserving diffeomor-phisms that are isomorphic to their inverses. In the jargon: the collection of diffeo-morphisms isomorphic to their inverses is “Π -hard.” The proof of the main theorem in this paper depends on background in two subjects,requiring the quotation of key results that would be prohibitive to prove. The actualconstruction itself—that is, the reduction F of the main theorem—is described inits entirety, along with the intuition behind these results.The paper heavily uses results proved in [9], [11], [12] and [13]. When used,the results are quoted, and informal intuition is given for the proofs. When specificnumbered lemmas, theorems and equations from [11] are referred to, the numberscorrespond to the arXiv version cited in the bibliography. Structure of the paper
The logical background required for the proof of Theo-rem 1 is minimal and the exposition is aimed at an audience with a basic workingknowledge of ergodic theory, in particular the Anosov-Katok method.Section 2 defines the odometer-based transformations, a large class of measurepreserving symbolic systems. These are built by iteratively concatenating wordswithout spacers. We then construct the reduction F O from the true Π statementsto the ergodic odometer-based transformations isomorphic to their inverse.Section 3 moves from symbolic dynamics to smooth dynamics. This proceeds intwo steps. The first step is to define a class of symbolic systems, the circular systems that are realizable as measure preserving diffeomorphisms of the two-torus. Thesecond step uses the Global Structure Theorem of [12], which shows that the categorywhose objects are odometer-based systems and whose morphisms are synchronousand anti-synchronous joinings is canonically isomorphic with the category whoseobjects are circular systems and whose objects are synchronous and anti-synchronous10oinings. Thus the odometer-based systems in the range of F O can be canonicallyassociated with symbolic shifts that are isomorphic to diffeomorphisms.Section 3.2 shows that different elements of the range of F ◦ F O are not isomor-phic, by showing that their Kronecker factors are different. Sections 3.3 discussesdiffeomorphisms of the torus and how to realize circular systems using method of Approximation by Conjugacy due to Anosov and Katok. Section 3.3 builds a prim-itive recursive map R from circular construction sequences to measure preservingdiffeomorphisms of T such that K c ∼ = R ( K c ).In section 3.4 we argue that the functor F defined in the Global StructureTheorem is itself a reduction when composed with F O . Hence composing R , F and F O gives a reduction F from the collection of true Π statements to the collectionof ergodic diffeomorphisms of the torus that are isomorphic to their inverse. Thiscompletes the proof of Theorem 1.The overall content of the paper is summarized by Figure 1. The reduction toodometer-based systems is F O , F is the functorial isomorphism, the realization assmooth transformations is R and the composition F is the reduction in Theorem 1.Figure 1: The reduction F . The Appendix
In the course of the proof of Theorem 1 various numerical pa-rameters are chosen with complex relationships. The are collected, explicated andshown to be coherent in Appendix A.Sections 2 and 3 of the body of the paper use certain standard notions andconstructions in ergodic theory and computability theory. A complete presentationis impossible, but for readers who want an general overview we present basic ideasfrom each subject as well exhibit explicit formulations of certain techniques used inthe paper.Appendix B is an overview of the logical background necessary for the proof ofthe theorem. It includes a basic description of Π formulas, a discussion of boundedquantifiers, how to express Goldbach’s conjecture as a Π formula and the definition11f “truth.” Appendix B.2 gives basic background on recursion theory, computablefunctions, and primitive recursion. Appendices B.2.2 and B.2.3 give background oneffectively computable functions.Appendix C gives background about ergodic theory and measure theory. It in-cludes the notion of a measurable dynamical system, the Koopman operator, and theergodic theorem. Appendix C.3 describes symbolic systems and gives the notationand basic definitions and conventions used in this paper. Appendix C.4 gives basicfacts about odometers and odometer-based systems. These include the eigenvaluesof the Koopman Operator associated to an odometer transformation and the canon-ical odometer factor associated with an odometer-based system. Appendix C.5 givesbasic definitions including the relationship between joinings and isomorphisms. Itdiscusses disintegrations and relatively independent products.Appendix D gives basic definitions of the space of C ∞ diffeomorphisms and givesan explicit construction of a smooth measure preserving near-transposition of adja-cent rectangles. The latter is a tool used in constructing the smooth permutations ofsubrectangles of the unit square. These permutations are the basic building blocksof the approximations to the diffeomorphisms in the reduction. The section verifiesthat these are recursive diffeomorphisms with recursive moduli of continuity andthat they can be given primitively recursively. Gaebler’s Theorem
The writing of this paper began as a collaboration betweenJ. Gaebler and the author with the goal of recording Foreman’s results establishingTheorem 1. Mathematically, Gaebler was concerned with understanding the foun-dational significance of Theorem 1. Though unable to finish this writing project,Gaebler established the following theorem in Reverse Mathematics:
Theorem (Gaebler’s Theorem) . Theorem 1 can be proven in the system ACA . This result will appear in a future paper [14].
Acknowledgements
The author has benefited from conversations with a largenumber of people. These include J. Avigad, T. Carlson, S. Friedman, M. Magidor,A. Nies, T. Slaman (who pointed out the analogy with Hilbert’s 10th problem),J. Steel, H. Towsner, B. Velickovic and others. B. Kra was generous with suggestionsfor the emphases of the paper and with help editing the introduction. B. Weiss, wasalways available and as helpful as usual. Finally my colleague A. Gorodetski wasindispensable for providing suggestions about how to edit the paper to make it moreaccessible to dynamicists.
In this section we prove the existence of the preliminary reduction F O .12 heorem 9. There is a primitive recursive function F O from the codes for Π -sentences to primitive recursive construction sequences for ergodic odometer basedtransformations such that:1. N is the code for a true statement if and only if F O ( N ) is the code for a con-struction sequence with limit T , where T is measure theoretically isomorphicto T − .2. For M (cid:54) = N , F O ( M ) is not isomorphic to F O ( N ) . Remark 10.
When discussing the construction of F O and F we will always havethe unstated assumption that the input N is a G¨odel number of a Π -statement.This is justified by remarking that, though formally the domain of F O (and soof F ) is the collection of N that are G¨odel numbers of Π -statements, the collectionof G¨odel numbers of Π -statements is primitive recursive. Theorem 1 is equivalentto asking that F be defined on all of N and output a code for the identity map whenthe input is an N that is not a G¨odel number of a Π -statement. Both Odometer Based and Circular symbolic systems are built using constructionsequences , a tool we now describe. They code cut-and-stack constructions and givea collection of words that constitute a clopen basis for the support of an invariantmeasure.Fix a non-empty alphabet Σ. If W is a collection of words in Σ, we will saythat W is uniquely readable if and only if whenever u, v, w ∈ W and uv = pws theneither: • p = ∅ and u = w or • s = ∅ and v = w .A consequence of unique readability is that an arbitrary infinite concatenation ofwords from W can be uniquely parsed into elements of W .Fix an alphabet Σ. A Construction Sequence is a sequence of collections ofuniquely readable words (cid:104)W n : n ∈ N (cid:105) with the properties that:1. Each word in W n is in the alphabet Σ.2. For each n all of the words in W n have the same length q n . The number ofwords in W n will be denoted s n .3. Each w ∈ W n occurs at least once as a subword of every w (cid:48) ∈ W n +1 .13. There is a summable sequence (cid:104) (cid:15) n : n ∈ N (cid:105) of positive numbers such that foreach n , every word w ∈ W n +1 can be uniquely parsed into segments u w u w . . . w l u l +1 (1)such that each w i ∈ W n , u i ∈ Σ < N and for this parsing (cid:80) i | u i | q n +1 < (cid:15) n +1 . (2)The segments u i in condition 1 are called the spacer or boundary portions of w . Theuniqueness requirement in clause 3 implies unique readability of each word in every W n .Let K be the collection of x ∈ Σ Z such that every finite contiguous subwordof x occurs inside some w ∈ W n . Then K is a closed shift-invariant subset of Σ Z that is compact if Σ is finite. The symbolic shift ( K , sh ) will be called the limit of (cid:104)W n : n ∈ N (cid:105) . Definition 11.
Let f ∈ K where K is built from a construction sequence (cid:104)W n : n ∈ N (cid:105) . Then by unique readability, for all n there is a unique w ∈ W n and a n ≤ < b n such that f (cid:22) [ a n , b n ) ∈ W n . This w is called the principal n -subword of f . If theprincipal n -subword of f lies on [ a n , b n ) we define r n ( f ) = − a n , the location of f (0) relative to the interval [ a n , b n ) .The construction sequences built in this paper are strongly uniform in that foreach n there is a number f n such that each word w ∈ W n occurs exactly f n times ineach word w (cid:48) ∈ W n +1 . It follows that ( K , sh ) is uniquely ergodic. We note that in definition 11 we must have b n − a n = q n . Notation
For a word w ∈ Σ < N we will write | w | for the length of w . Inverses and reversals If K is a symbolic shift built from a construction sequence (cid:104)W n : n ∈ N (cid:105) then we can consider its inverse in two ways. The first is ( K , Sh − ).The second, which we call Rev ( K ) is the system built from the construction sequence (cid:104) Rev ( W n ) : n ∈ N (cid:105) where Rev ( W n ) is the collection of reversed words from W n :if w ∈ W n then w written backwards belongs to Rev ( W n ). Clearly ( K , Sh − ) isisomorphic to ( Rev ( K ) , sh ) and we will use both conventions depending on context. Odometer Based construction sequences
A construction sequence with W =Σ and built without spacers is called an odometer-based construction sequence.For odometer-based sequences, Clause 3 of the definition of Construction Sequence implies that for odometer based systems W n +1 ⊆ W k n n for some sequence (cid:104) k n : n ∈ N (cid:105) of natural numbers with k n ≥
2. Hence |W n +1 | ≤ |W n | k n . In the special case14f odometer sequences we write the length of words in W n as K n . We note that K n = (cid:81) n − m =0 k m .The sequence (cid:104) k n : n ∈ N (cid:105) determines an odometer transformation with domainthe compact space O = def (cid:89) n Z k n . The space O is naturally a monothetic compact abelian group. We will denotethe group element (1 , , , , . . . ) by ¯1, and the result of adding ¯1 to itself j timesby ¯ j . There is a natural map of O given by O ( x ) = x + ¯1. Then O is a topologi-cally minimal, uniquely ergodic invertible homeomorphism of O that preserves Haarmeasure. The map x (cid:55)→ − x is an isomorphism of O with O − . (See Appendix C.4and [9] for more background.)Odometer transformations are characterized by their Koopman operators. Theyare discrete spectrum and the group of eigenvalues is generated by the K n -th rootsof unity. The odometer factor If K is built from an odometer-based construction sequenceand the principal n -subword of f sits at [ − a n , b n ) then the sequence (cid:104) a n : n ∈ N (cid:105) gives a well defined member π O ( f ) of O = (cid:81) i Z k i . It is easy to verify that the map f (cid:55)→ π O ( f ) is a factor map.A measure preserving transformation is odometer-based if it is finite entropy,ergodic and has an odometer factor. It is shown in [10] that every odometer-basedtransformation is isomorphic to a symbolic shift with an odometer-based construc-tion sequence. Fix an odometer based construction sequence (cid:104)W n : n ∈ N (cid:105) . If Q is an equivalencerelation on W n , then elements of K can be viewed as determining sequences ofequivalence classes. More precisely if Σ ∗ is the alphabet consisting of classes W n / Q we can consider the collection W ∗ n of words of length K n that are constantly equal toan element of Σ ∗ . Let m > n . Then for some K , the words in W m are concatenationsof K -words from W n . Viewed this way, the words in W m determine a sequence of K many elements of W ∗ n . Concatenating them we get a word of length K m that isconstant on contiguous blocks of length K n . Let W ∗ m be the collection of words inthe alphabet Σ ∗ arising this way. There is a clear projection map π : W m → W ∗ m that sends two words in W m to the same word in W ∗ m if they induce the samesequence of Q -classes.Equivalently define the diagonal equivalence relation Q K on W Kn by setting w w . . . w K − ∼ w (cid:48) w (cid:48) . . . w (cid:48) K −
15f and only if for all i, w i ∼ Q w (cid:48) i . Then for two words w, w (cid:48) ∈ W m , π ( w ) = π ( w (cid:48) ) ifand only if w ∼ Q K w (cid:48) . Similarly let w ∈ ( W n /Q ) K and w (cid:48) ∈ W Km . Then w (cid:48) is a substitution instance of w if and only if w (cid:48) = w w · · · w K − and w = [ w ] Q [ w ] Q · · · [ w K − ] Q . The sequence (cid:104)W ∗ m : m ≥ n (cid:105) determines a well-defined odometer-based construc-tion sequence in the alphabet Σ ∗ . If we define K Q to be the limit of (cid:104)W ∗ m : m ≥ n (cid:105) then there is a canonical factor map π Q : K → K Q .We now discuss how this factor map behaves with inverse transformation. Sup-pose that Z acts freely on Σ ∗ = W n / Q . Then for all K we can extend this actionto (Σ ∗ ) K by the skew-diagonal action. Suppose that g is the generator of Z . Define g · ([ w ] Q [ w ] Q . . . [ w K − ] Q ) = g · [ w K − ] Q g · [ w K − ] . . . g · [ w ] . Assume that W ∗ m is closed under the skew-diagonal action. Let w = [ w ][ w ][ w K − ] ∈ W ∗ m . Then we can apply g pointwise to the [ w i ]; i.e. the diagonal action. Since W ∗ m isclosed under the skew-diagonal action, the word g [ w ] g [ w ] . . . g [ w K − ] ∈ Rev ( W ∗ m ). Lemma 12.
Suppose for all m > n, W ∗ m is closed under the skew-diagonal actionof g . Then K Q ∼ = Rev ( K Q ) and the isomorphism takes an f ∈ K Q with associatedodometer sequence x to an element of Rev ( K Q ) determined by the diagonal actionthat has associated odometer sequence − x . (cid:96) The sequence (cid:104)
Rev ( W ∗ m ) : m ≥ n (cid:105) is a construction sequence for Rev ( K Q ). Themap [ w ][ w ] . . . [ w K − ] (cid:55)→ g [ w ] g [ w ] . . . g [ w K − ] ∈ Rev ( W ∗ m )is an invertible shift-equivariant map defined on the construction sequences for K Q and Rev ( K Q ) and hence defines an invertible graph joining η g from K Q to Rev ( K Q ) (cid:97) We note that the graph joining η g does not depend on which elements of W n areidentified by Q . Moreover to recover Rev ( K ) from Rev ( K Q ) one substitutes theappropriate reverse words Rev ( w ) into a Q -class C . Frequently the graph joining η g of K Q with Rev ( K Q ) does not come from a graph joining of K with Rev ( K ).In the construction in [9], which we modify in this paper, this process is iterated:there is an equivalence relation Q on W n and another equivalence relation Q on W n with n < n and Q a refinement of the product equivalence relation Q K (for We note in passing that being closed under the skew diagonal action does not imply that W m / Q n is closed under reverses. K ). There will be two copies of Z generated by g and g with g acting freely on W n / Q and g acting freely on W n / Q .For i = 1 , W m / ( Q i ) K by ( W ∗ m ) i . We build two construction sequencesconsisting of collections of words made up of equivalence classes (cid:104) ( W ∗ m ) : m ≥ n (cid:105) and (cid:104) ( W ∗ m ) : m ≥ n (cid:105) which we assume are closed under the skew-diagonal actionsof g and g respectively. Let K be the limit of (cid:104) ( W ∗ m ) : m ≥ n (cid:105) and K the limitof (cid:104) ( W ∗ m ) : m ≥ n (cid:105) .Then we get a tower KK K (cid:63)(cid:63) Suppose the g action on Q is subordinate to the g action on W n / ( Q ) K ; thatis, whenever C and C are classes of W n / ( Q ) K and W n / Q and C ⊆ C , then g C ⊆ g C .Then the various projection maps between K , K Q and K Q commute with theshift and the joining η g of K Q × Rev ( K Q ) extends the joining η g of K Q × Rev ( K Q ). Given an infinite sequence of equivalence relations Q i , the associatedjoinings cohere into an invertible graph joining of K with Rev ( K ) if and only if the σ -algebras associated with the K Q i generate the measure algebra on K . Diagonal vs Skew-diagonal actions.
Since (cid:121) n extends to both the diagonaland skew-diagonal actions, we summarize the distinct rolls: • The skew-diagonal actions give closure properties on W m / Q mn = ( W ∗ m ) n . • Because of this closure the diagonal action gives an isomorphism between K n and Rev ( K ) n . This approximates a potential isomorphism from K to Rev ( K ). The construction of the first reduction F O closely parallels the construction in [9]and we refer the reader to that paper for details of claims made here. For each N theroutine F O ( N ) inductively builds an odometer construction sequence (cid:104)W n : n ∈ N (cid:105) in the alphabet Σ = { , } with W n +1 ⊆ W k n n . During the construction we willaccumulate inductive numerical requirements. Some, such as the (cid:15) n ’s and the ε n ’sare positive numbers that go to zero rapidly. Some, such as the k n ’s and l n ’s aresequences of natural numbers that go to infinity. These numbers depend on N , sowhen necessary we will write W n ( N ), (cid:15) n ( N ), k n ( N ), l n ( N ) and so forth. However17or notational simplicity we will drop the N whenever it is clear from context. Atstage n in the algorithm F ( N ) for building W n ( N ), for M < N F can recursivelyrefer to objects build by F ( M ) at stages ≤ n . For example F ( N ) can assume that k n ( N −
1) is known.These sequences of numbers are defined inductively and have complex relation-ships, requiring some verification that they are consistent and can be chosen primi-tively recursively. That they are consistent is the content of section 10 of [11]. Thatthey can be chosen primitively recursively involves a routine review of the argumentsin that paper. For completeness this is done in Appendix A.
Numerical Requirement A s n = 2 ( n +1) e ( n − for an increasing sequence of nat-ural numbers e ( n ).The construction will use the following auxiliary objects and their properties:1. A sequence of equivalence relations (cid:104)Q n : 1 ≤ n < ∞(cid:105) . Each Q n is anequivalence relation on W n , hence gives a factor K n of K .2. The equivalence relation Q n +1 refines the product equivalence relation ( Q n ) k n on W k n n .3. The sub- σ -algebra H n of B ( K ) corresponding to K n . In the construction here,as with the original construction, (cid:83) n H n will generate B ( K ) modulo the setsof measure zero with respect to the unique shift-invariant measure µ . (This isLemma 14 which uses specification Q4.)We denote the sub- σ -algebra of B ( K ) corresponding to the odometer fact by H . Because the odometer factor sits in side each K n , H ⊆ H n for all n .4. A system of free Z actions (cid:121) n on W n / Q n for n < Ω. Denote the generatorof Z corresponding to (cid:121) n as g n .Suppose that n < m . As in section 2.2, the words in W m are concatenations of K = K m /K n -many words from W n . Hence the product equivalence relation ( Q n ) K gives an equivalence relation on W m , which we call Q mn . We will denote W m / Q mn by ( W ∗ m ) n . The Z actions have the following properties: • ( W ∗ m ) n is closed under the skew-diagonal action of g n . • If n + 1 < Ω, then the g n +1 action is subordinate to the g n action. • We let (cid:121) n be the diagonal action of g n on K n . Since ( W ∗ m ) n is closed under theskew-diagonal action, (cid:121) n can be viewed as mapping ( W ∗ m ) n to Rev (( W ∗ m ) n ).As described in section 2.2, for n < Ω, (cid:121) n canonically creates an isomorphismbetween K n and K − n that induces the map x (cid:55)→ − x on the odometer factor.18estating this: if the action (cid:121) n is non-trivial, then it induces a graph joining η n of H n with ( H n ) − that projects to the map x (cid:55)→ − x on the odometer factor.Assuming n + 1 < Ω, the action (cid:121) n +1 is subordinate to (cid:121) n , the joining η n +1 projects to the joining η n . If Ω = ∞ , since the (cid:83) n H n will generate B ( K ), the η n ’swill consequently cohere into a conjugacy of T with T − .Lemmas 26 and 27 of [9] formalize this and show the following conclusion. Lemma 13.
Suppose
Ω = ∞ . Then there is a measure isomorphism η of K with K − such that for all n ∈ N , η induces an isomorphism η n : K n → K n that coincideswith the graph joining determined by the action of the generator for (cid:121) n on K n . The construction is arranged so that if the number Ω in clause 3 of the descriptionof the objects is finite, then K (cid:54)∼ = K − . This is done by making the sequences ofequivalence classes of elements of ( W ∗ m ) n = W m / Q mn essentially independent of theirreversals subject to the conditions described above. The specifications given laterin this section make this precise. F O . The algorithm for the reduction F O is diagrammed in Figure 2.Given N , F O determines the Π formula coded by N : ϕ N = ∀ z ∀ z . . . ∀ z m ϕ ( z , z , . . . z m ) . The function F then uses the formula to generate a computational routine R ϕ that recursively computes the objects W n ( N ) , Q n ( N ) and (cid:121) n ( N ) (as well as thevarious numerical parameters that are involved in the construction). Here is what R ϕ does. The routine R ϕ
1. Fixes a computable enumeration of all m -tuples (cid:104) (cid:126)z n = ( z , . . . z m ) n : n ∈ N (cid:105) ofnatural numbers2. On input n , R ϕ initializes i = 0, sets W = { , } , Q the trivial equivalencerelation with one class and the action (cid:121) the trivial action.3. For i < n , R ϕ :(a) builds W i +1 , Q i +1 ,(b) computes (cid:104) (cid:126)z j : 0 ≤ j ≤ i (cid:105) ,(c) Asks: “Is ϕ N ( (cid:126)z j ) true for all 0 ≤ j ≤ i ?”Since ϕ N has no unbounded quantifiers, this question is primitive recur-sive. 19nput n , andset i to 0. Initialize W ( M ), G ( M ), and Q ( M ) for M ≤ N .Is i < n ? Return W n ( N ), Q nn ( N ), (cid:121) n ( N )Set G i +1 ( N )to Z / Z and definethe action (cid:121) i +1 ( N ). Build W i +1 ( M )and Q i +1 j ( M )for 0 ≤ j ≤ i and M ≤ N Set i = i + 1 Is ϕ N ( (cid:126)z j )true for all0 ≤ j ≤ i ? Calculate (cid:126)z j for 0 ≤ j ≤ i .Set G i +1 ( N )to { e } , thetrivial group. yesno noyesFigure 2: The algorithm R ϕ = F O ( N ).20d) If yes , R ϕ builds the action (cid:121) i +1 (e) If no , R ϕ makes the (cid:121) i +1 trivial. (Note that if i is the first integer inthis case, then Ω will equal i + 1.)4. When i = n , R ϕ returns W n . We describe the construction sequence, the equivalence relations and the actions.To start we choose a prime number P > (cid:104) P N : N > (cid:105) enumerate the prime numbers bigger than P .For the construction sequence corresponding to F O ( N ), words in W have length P N . The words in W n will have length K n = P N (cid:96) for some (cid:96) chosen large enoughas specified below. The K n ’s will be increasing and K m divides K n for m < n .Let k n = K n +1 /K n . Thus k n is a large power of 2 and each word in W n +1 is aconcatenation of k n many words from W n . The number of words in W n is s n . Werequire that s n divides s n +1 and s n is a power of 2 that goes to goes to infinityquickly. Since W n +1 ⊆ W k n n this induces lower bounds on the growth of the k n ’s.The requirements described here are simpler than those in [9] as modified in[11], and the “specifications” used there are appropriately simplified or omitted ifnot relevant to this proof. The construction carries along numerical parameters (cid:104) (cid:15) n (cid:105) , (cid:104) k n (cid:105) , (cid:104) K n (cid:105) , (cid:104) s n (cid:105) , (cid:104) Q n (cid:105) , (cid:104) C n (cid:105) , and (cid:104) e ( n ) (cid:105) . (Showing that the various coefficients arecompatible and primitively recursively computable appears in Appends A.)As an aid to the reader we use the analogous labels for the simplified specifica-tions as those that appear in [11]. Q4 For n ≥
1, any two W n -words in the same Q n class agree on an initial segmentof proportion at least (1 − (cid:15) n ). Q6.
As a relation on W n +1 , for 1 ≤ s ≤ n + 1, Q n +1 s refines Q n +1 s − and each Q ns − class contains 2 e ( n ) many Q ns classes.The point of Q4 is that the Q n classes approximate words in W n by specifyingarbitrarily long proportions of the words. A consequence of this is: Lemma 14. (cid:83) n H n generates the measure algebra of K . (cid:96) This is proved in Proposition 23 of [9]. (cid:97)
Thus Q4 is the justification for Assertion 3 of Section 2.3.We now turn to the joining specifications. These are counting requirements thatdetermine the joining structure. The joining specifications we present here are morecomplicated than strictly necessary for the simplified construction in this paper, but21e present them as appear in [9] in order to be able to directly quote the theoremsproved there. We note that specification J10.1 is a strengthening of J10 in [9].Suppose that u and v are elements of W n +1 ∪ Rev ( W n +1 ) and ( u (cid:48) , v (cid:48) ) an orderedpair from W n ∪ Rev ( W n ). Suppose that u and v are in positions shifted relative toeach other by t units. Then an occurrence of ( u (cid:48) , v (cid:48) ) in ( sh t ( u ) , v ) is a t (cid:48) such that u (cid:48) occurs in u starting at t + t (cid:48) and in v starting at t (cid:48) . If X is an alphabet and W isa collection of words in X , and u ∈ W ∪ Rev ( W ) we say that u has forward parity if u ∈ W and reverse parity if u ∈ Rev ( W ).By specification Q4 no word in W n +1 belongs to Rev ( W n +1 ), so parity is well-defined and unique. However the words in ( W ∗ n ) i may belong to Rev (( W ∗ n ) i ) andwe view those words as having both parities. J10.1
Let u and v be elements of W n +1 ∪ Rev ( W n +1 ). Let 1 ≤ t < (1 − (cid:15) n )( k n ).Let j be a number between (cid:15) n k n and k n − t . Then for each pair u (cid:48) , v (cid:48) ∈W n ∪ Rev ( W n ) such that u (cid:48) has the same parity as u and v (cid:48) has the sameparity as v , let r ( u (cid:48) , v (cid:48) ) be the number of j < j such that ( u (cid:48) , v (cid:48) ) occurs in( sh tK n ( u ) , v ) in the j · K n -th position in their overlap. Then (cid:12)(cid:12)(cid:12)(cid:12) r ( u (cid:48) , v (cid:48) ) j − s n (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) n . For fixed n and s , let Q ns = | ( W ∗ n ) s | and C ns be the number of equivalent elementsin each block of the partition W n / Q ns . J11
Suppose that u ∈ W n +1 and v ∈ W n +1 ∪ Rev ( W n +1 ). We let s = s ( u, v ) bethe maximal i < Ω such that [ u ] i and [ v ] i are in the same (cid:121) i -orbit. Let g = g i and ( u (cid:48) , v (cid:48) ) ∈ W n × ( W n ∪ Rev ( W n )) be such that g [ u (cid:48) ] s = [ v (cid:48) ] s . Let r ( u (cid:48) , v (cid:48) )be the number of occurrences of ( u (cid:48) , v (cid:48) ) in ( u, v ). Then: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r ( u (cid:48) , v (cid:48) ) k n − Q ns (cid:18) C ns (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) n . The next assumption is a strengthening of a special case of J11.
J11.1
Suppose that u ∈ W n +1 and v ∈ W n +1 ∪ Rev ( W n +1 ) and [ u ] not in the (cid:121) -orbit of [ v ] . Let j be a number between (cid:15) n k n and k n . Suppose that I is either an initial or a tail segment of the interval { , , . . . K n +1 − } havinglength j K n . Then for each pair u (cid:48) , v (cid:48) ∈ W n ∪ Rev ( W n ) such that u (cid:48) has thesame parity as u and v (cid:48) has the same parity as v , let r ( u (cid:48) , v (cid:48) ) be the numberof occurrences of ( u (cid:48) , v (cid:48) ) in ( u (cid:22) I, v (cid:22) I ). Then: (cid:12)(cid:12)(cid:12)(cid:12) r ( u (cid:48) , v (cid:48) ) j − s n (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) n . In the language of J11: s ( u, v ) = 0, Q n = 1 and C n = s n . Remark 15.
We note that specification J10.1 implies unique readability of thewords in W n +1 . This follows by induction on n . If the words in W n +1 were notuniquely readable then we would have u, v, w ∈ W n with uv = pws and neither p nor s empty. But the one of u or v would have to overlap either an initial segmentor a tail segment of w of length K n +1 / . Suppose it is an initial segment of w anda tail segment u . On this tail segment the n -subwords would have to agree exactlywith the n -subwords of an initial segment of w . But this contradicts J . . Suppose we have built a collection of words (cid:104)W n : n ∈ N (cid:105) , equivalence relations (cid:104)Q n : n ∈ N (cid:105) and actions (cid:104) (cid:121) n : n ∈ N (cid:105) satisfying the properties described then wecan cite the following results occurring in [9]. Fix a transformation T built withthe construction sequence (cid:104)W n : n ∈ N (cid:105) . Recall that if (cid:121) n is non-trivial then thegenerator g n (cid:54) = 0 induces an invertible graph joining η n of K n with K − n . We quotethe following results of [9], referencing their numbers in that paper. Theorem 13 and Proposition 32
Suppose that η is an ergodic joining of T with T − that is not a relatively independent joining over the odometer factor.Then η (cid:22) H × H is supported on the graph of some ¯ j -shift of the odometerfactor. Proposition 37 If η is an ergodic joining of K with K − , then exactly one of thefollowing holds:1. Ω < ∞ and for some n ≤ Ω, j ∈ Z and some η n , η is the relativelyindependent joining of K with K − over the joining η n ◦ (1 , sh − j ) of K n × K − n .2. Ω = ∞ and for some j , all n the projection of η to a joining on K n × K − n is of the form η n ◦ (1 , sh − j )If Ω = ∞ , since the H n ’s generate, η is an invertible graph joining of K with K − . In both cases the projection of η n to a joining of the odometer factorwith itself concentrates on the map x (cid:55)→ − x .Thus it follows that:1. If K ∼ = K − then Ω = ∞ . In particular if K ∼ = K − , then the Π statement ϕ N is true.2. The projection of η n ◦ (1 , sh − j ) to the odometer is of the form x (cid:55)→ − x − j .3. Similarly the projection of η ◦ (1 , sh − j ) to the odometer is of the form x (cid:55)→− x − j . 23lause 2 of Theorem 9 requires that if M (cid:54) = N are different codes for Π sentencesthen the transformation F O ( M ) is not isomorphic to F O ( N ). This is clear becausethe odometer sequence for F O ( M ) consists of k ’s whose prime factors are 2 and P M ,while the odometer sequence for F O ( N ) has k ’s whose prime factors are 2 and P N .Since P M (cid:54) = P M , the odometer factors are not isomorphic.Corollary 33 of [9] implies that the Kronecker factor of each F O ( N ) is the odome-ter factor. Since any isomorphism ϕ between F O ( M ) with F O ( N ) must induce anisomorphism of the Kronecker factors, ϕ has to induce an isomorphism of the cor-responding odometer factors, yielding a contradiction. (See Corollary 57 in Ap-pendix C)To finish the proof of Theorem 9 we must show that the words, equivalencerelations and actions can be built primitively recursively. To finish the proof of Theorem 9 the words W n ( N ), the equivalence relations Q n ( N )and actions (cid:121) n ( N ) must be constructed and it must be verified that the construc-tion is primitive recursive. Note:
Formally we are just constructing actions (cid:121) n for n < Ω. However fornotational convenience, when constructing the words at stage n + 1, we will write (cid:121) i when Ω ≤ i < n + 1 with the understanding that it is the trivial identity action.The collections of words W n are built probabilistically. A finitary version of lawof large numbers shows that there are primitive recursive bounds on the length ofthe words in a collection with the necessary properties. The actual collection ofwords can then be found with an exhaustive search of collections of words of thatlength, showing that the entire construction is primitive recursive. Structure of the induction.
The collections of words W n are built by inductionon n . For n ≥ W n +1 are built by iteratively substituting words into k n -sequences of classes Q ni , by induction on i ≤ n . We will adapt the notation ofsection 2.2.The length K of words in W will be a large prime number P N . To pass fromstage n to n + 1, one is required to build the words W n +1 , the equivalence relation Q n +1 and, if n + 1 < Ω the action (cid:121) n +1 . The length K n +1 of the words will be2 (cid:96) · K n for an (cid:96) taken large enough.Suppose we have already chosen k n and it is a large power of 2. Then ( Q ni ) k n for 0 ≤ i ≤ n give us a hierarchy of equivalence relations of potential words asdescribed in Section 2.2 as well as the diagonal and skew-diagonal actions of (cid:121) i for i < min( n, Ω). 24 emark 16.
The construction of W n +1 is top-down. We construct the ( W ∗ n +1 ) i = W n +1 / Q n +1 i by induction on i before we construct W n +1 . The equivalence rela-tions get more refined as i increases, so each step gives more information about W n +1 . Having built ( W ∗ n +1 ) n , an additional step constructs creates W n +1 and theequivalence relation Q n +1 n +1 . Start with i = 0. Then W n / Q n has one element, a string of length K n witha single letter. Let ( W ∗ n +1 ) be the single element consisting of strings of length k n · K n in that single letter.Each element of ( W ∗ n +1 ) is built by substituting k n elements of ( W ∗ n ) —each ofwhich is a contiguous block of length K n —into ( W ∗ n +1 ) . We continue this processinductively, ultimately arriving at ( W ∗ n +1 ) n .The elements X being substituted The result of the substitutioninto previous words ( W ∗ n +1 ) ( W ∗ n ) ( W ∗ n +1 ) ( W ∗ n ) ( W ∗ n +1 ) ... ...( W ∗ n ) n ( W ∗ n +1 ) n The result of this induction is a sequence of elements of W n / Q n of length k n ∗ K n ,that is constant on blocks of length K n . We must finish by substituting elements of W n into the W n / Q n -classes to get W n +1 and defining Q n +1 . A step in the induction on i . Fix an i and view elements ( W ∗ n +1 ) i as k n -sequences C C . . . C k n − of elements of ( W ∗ n ) i . Since Q i +1 refines the diagonalequivalence relation ( Q i ) K i +1 /K i , ( Q ni +1 ) k n refines ( Q ni ) k n . Inside each Q ni class C j , one can choose a Q ni +1 class C (cid:48) j ∈ ( W ∗ n ) i +1 . Concatenating these to get C (cid:48) C (cid:48) . . . C (cid:48) k n − we create an element of ( W ∗ n +1 ) i +1 . We do the construction so thatresult is closed under the skew diagonal action of (cid:121) i +1 . Remark 17.
Following section 2.2, elements of ( W ∗ i +1 ) i +1 are constant sequences oflength K i +1 . Thus the concatenation C (cid:48) C (cid:48) . . . C (cid:48) k n − is a sequence of k n ∗ ( K n /K i +1 ) many contiguous constant blocks of length K i +1 . We now describe how these choices are made. Our discussion is aimed at thecase where n + 1 < Ω, for n + 1 ≥ Ω take (cid:121) n +1 to be the trivial action. Fix acandidate k for k n . View { rev } as acting on ( W n / Q ni ) k = (( W ∗ n ) i ) k . Together, theskew-diagonal action of (cid:121) i and { rev } generate an action on ( W n / Q ni ) k . Let R i bea set of representatives of each orbit of this action. Fix the number E of i + 1-classes25esired inside each i -class. Consider X i = (cid:89) r ∈ R i E − (cid:89) q =0 S ( r, q ) , (3)where S ( r, q ) is the collection of all substitution instances of Q ni +1 classes into r . More explicitly, if r = C C . . . C k − where C j ∈ W n / Q ni . Let C ∗ j = { C (cid:48) : C (cid:48) ⊆ C j and C (cid:48) ∈ W n / Q ni +1 } . For each 0 ≤ q ≤ E −
1, let S ( r, q ) = k − (cid:89) j =0 C ∗ j . Fix an r ∈ R i . The every element W of (cid:81) E − q =0 S ( r, q ) can be viewed as a collectionof E many words of length k in the language ( W ∗ n ) i +1 whose Q ni classes form r . Eachof these E many words can be copied by the (cid:121) i +1 action. If w is such a word, andis a substitution instance of r then (cid:121) i +1 ( w ) is a substitution instance of (cid:121) i ( r ).So comparing elements of W (and their shifts) is the same as comparing potentialwords in ( W ∗ n +1 ) i +1 . The action of (cid:121) i +1 preserves the frequencies of occurrences ofwords inWe work with X i because it can be viewed as a discrete measure space with thecounting measure. The objects being counted in the various specifications corre-spond to random variables on this measure space. Definition 18. If (cid:104) w r,q : r ∈ R i , ≤ q < E (cid:105) is the collection of words built usingthe Substitution Lemma passing from stage i to stage i + 1 , the ( W ∗ n +1 ) i +1 is theclosure of { w r,q : r ∈ R i , ≤ q < E } under the skew-diagonal action of (cid:121) i . Example 19. If C ⊆ C j , D ⊆ C j (cid:48) are substitution instances, we have the in-dependent random variables X r,q,j , X r (cid:48) ,q (cid:48) ,j (cid:48) taking value 1 at points (cid:126)x ∈ X i where x ( r, q, j ) = C and x ( r (cid:48) , q (cid:48) , j (cid:48) ) = D , respectively. The event that C occurs in X i inthe q th word in position j and D occurs in r (cid:48) in the ( q (cid:48) ) th word in position j (cid:48) is theevent that both X r (cid:48) ,q (cid:48) ,j (cid:48) = 1 and X r,q,j = 1 . If each i -class has p elements then theprobability that both X r (cid:48) ,q (cid:48) ,j (cid:48) = 1 and X r,q,j = 1 is /p . The strong law of large numbers tells us that the collection of points in each X i that do not satisfy the specifications (as they are coded in the conclusion of theSubstitution Lemma) goes to zero exponentially fast in k . As k grows, the numberof requirements to satisfy the Substitution Lemma grows linearly. Hoeffding’s in-equality (Theorem 21 below) says that the probabilities stabilize exponentially fast.The Substitution Lemma follows. Note that q is a dummy index variable here. n more detail: The word construction proceeds by first getting a very closeapproximation to what is desired and then finishing the approximations to exactlysatisfy the requirements. These two steps correspond to Proposition 43 and Lemma41 of [9].The general setup for the Substitution Lemma (Proposition 20) at stage n + 1is as follows: • An alphabet X and an equivalence relation Q on X , with Q classes each ofcardinality C . • A collection of words
W ⊆ ( X/ Q ) k for some k . • Groups
G, H with generators g, h that are either Z or the trivial group. If H = Z then G = Z . • If G = Z then we have a free action G (cid:121) X/Q and if H = Z we also havea free action H (cid:121) X . Thus the skew-diagonal actions of G on ( X/Q ) k and H on X k are well-defined. If either group is trivial, then the correspondingactions are trivial. • The H (cid:121) X action is subordinate to G (cid:121) X/Q action via ρ . • Constants (cid:15) a , (cid:15) b ∈ (0 ,
1) such that (cid:15) b < (cid:15) a / | X | . • A constant E determining the number of substitution instances desired foreach Q class. • If u, v, w, w (cid:48) are words in the alphabet X , then r ( u, v, sh i ( w ) , w (cid:48) ) is the numberof j such that u occurs in w starting at j + i and v occurs in w (cid:48) starting at j . Similarly if u, w are words in the alphabet X , the r ( u, w ) is the number ofoccurrences of u in w .A special case of the Substitution Lemma (Proposition 63 in [9]) is: Proposition 20 (Substitution Lemma) . Let
E > be an even number. There is alower bound k lb depending on ( (cid:15) b , (cid:15) a , Q, C, W, E ) such that for all numbers k ≥ k lb and all symmetric W ⊆ ( X/ Q ) k with cardinality W that are closed under the skew-diagonal action of G and Rev () , if for all i with ≤ i ≤ (1 − (cid:15) b ) k , u, v ∈ X/ Q and w, w (cid:48) ∈ W : (cid:12)(cid:12)(cid:12)(cid:12) r ( u, v, sh i ( w ) , w (cid:48) ) k − i − Q (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) b (4) and each u ∈ X/ Q occurs with frequency /Q in each w ∈ W , then there is a collection of words S ⊆ X k consisting of substitution instancesof W k such that if W (cid:48) = HS ∪ Rev ( HS ) we have: H is acting on X k by the skew-diagonal action. . Every element of W (cid:48) is a substitution instance of an element of W and eachelement of W has exactly E many substitution instances of words in W (cid:48) .2. For each x ∈ X and each w ∈ W (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) r ( x, w ) k − | X | (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) a (5) i.e., the frequency of x in w is within (cid:15) a of / | X | .3. If w , w ∈ S ∪ Rev ( S ) with [ w ] Q = [ w ] Q and w / ∈ H w and x, y ∈ X with [ x ] = [ y ] . Then for h ∈ H : (cid:12)(cid:12)(cid:12)(cid:12) r ( x, y, w , hw ) k − Q · C (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) a . (6)
4. Let i be a number with ≤ i ≤ (1 − (cid:15) a ) k and j be a number between (cid:15) a k/ and k − i , x, y ∈ X , w , w ∈ W (cid:48) ∪ Rev ( W (cid:48) ) , let r ( x, y ) be the number of j < j such that ( x, y ) occurs in ( sh i ( w ) , w ) in the j th position. Then (cid:12)(cid:12)(cid:12)(cid:12) r ( x, y ) j − | X | (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) a . (7)
5. For all x, y ∈ X and all w , w ∈ W (cid:48) ∪ Rev ( W (cid:48) ) with different H orbits, (cid:12)(cid:12)(cid:12)(cid:12) r ([ x ] Q , [ y ] Q , [ w ] Q , [ w ] Q ) k − c (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) b (8) implies that, (cid:12)(cid:12)(cid:12)(cid:12) r ( x, y, w , w ) k − cC (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) a . (9)We remark again that the Law of Large numbers implies that conclusions 1-5 holdfor almost all infinite sequences. For example if you perform i.i.d. substitutions ofelements of X to create a typical infinite sequence (cid:126)w , then the density of occurrencesof a given x in (cid:126)w will be 1 / | x | . The Hoeffding inequality says that the finitaryapproximations to this conclusion converge exponentially fast. As a result, for largeenough k it is possible to satisfy conclusions 1-5 with very high probability.Another remark is that at each stage we start with a collection of words W closedunder reversals and produce another collection of words W (cid:48) closed under reversals. While there are typographical errors in the statement of this item in [9], the proof given thereyields the correct statement which is inequality 6. Similarly, conclusion 4 has been strengthenedslightly here in a way that does not materially change the proof. he sequence e ( n ) . We will have a sequence e ( n ) such that for n ≥ s n +1 =2 ( n +2) e ( n ) that satisfies some growth conditions. (See Inherited Requirement 2 and
Inherited Requirement 3 in Appendix A and Figure 5 for an explicit statement ofthese conditions.) To initialize the construction we take e (0) = 2. Finding k n We now use Proposition 20 to build the collections of words. We willapply it with E = 2 e ( n ) except in one instance where we apply it with E = 2 e ( n ) .To start the inductive construction, we take P to be large enough to apply theSubstitution Lemma Q the trivial equivalence relation and W = Σ = { , } . For N >
0, since P N ≥ P , P N can also be used for k ( N ) to initialize the constructionas described below with n = 0.We then choose k n large enough to allow n + 1 successive Lemma 20-style sub-stitutions for E = 2 e ( n ) corresponding to the equivalence relations Q ni for 1 ≤ i ≤ n together with a final substitution of the letters in the base alphabet Σ to produce W n +1 . (This is a total of n + 2 substitutions.)More explicitly, note that each of the n + 2 applications of the SubstitutionLemma for the various Q i with E = 2 e ( n ) and (cid:15) a = (cid:15) n /
100 and the finishing lemmaproduces a lower bound k i lb .The following will be important later in the paper: Numerical Requirement B
Let k n ( N −
1) be the k n corresponding to the reduc-tion F O ( N −
1) and k n ( N ) be the k n corresponding to the reduction F O ( N )and k n ( N ). Then k n ( N ) ≥ k n ( N −
1) (10)Choose a large power of two k Max > max { k lb , k lb , . . . , k n lb , k n ( N − } , ensuring that it be sufficiently large that 2 − k Max < (cid:15) n . Then, set k n = k Max ∗ s n . (11)Since k n is of this form and s n is a power of 2, this ensures that K n +1 = P N · (cid:96) for a large (cid:96) . By increasing k Max if necessary we can also assume1. 1 /k n < (cid:15) n / s n +1 ≤ s k n n . Building W n +1 / Q n +1 i for i ≤ n : This is done by applying the SubstitutionLemma n times to pass from ( W ∗ n +1 ) successively to ( W ∗ n +1 ) n . At each i < n we substitute 2 e ( n ) many elements of ( W ∗ n +1 ) i +1 into each element of ( W ∗ n +1 ) i .29 ompleting W n +1 : Having constructed W n +1 / Q n it remains to construct W n +1 , Q n +1 and the action (cid:121) n +1 . The latter is only relevant if n + 1 < Ω.We must ensure that the resulting collection of words satisfy Q4 and Q6. Thisis accomplished by constructing two collections of words, the stems and the tails. Start by rewriting k Max as ( k Max − k Max ) + k Max . • The tails:
To build the tails, which have length k Max s n K n , we use Lemma20, with X = W n and Q = Q n to build 2 e ( n ) many substitution instances ineach Q n +1 n -class C of the final k Max s n portion of each word in ( W ∗ n +1 ) n . Wecall these the tails corresponding to C . • The stems:
The stems have length ( k Max − k Max ) s n K n . We use Lemma 20,again with X = W n and Q = Q n , to create 2 e ( n ) many substitution instancesin each initial segment of a ( W ∗ n +1 ) n -word of length k Max − k Max . We callthese the stems corresponding to the initial segments of the Q n +1 n -class C ofthis word.The words in W n +1 are built one Q n +1 n class at a time. Fix such a class C . Then C has 2 e ( n ) many stems in the first k Max − k Max and 2 e ( n ) many tails in the finalsegment of length k Max . Pair each stem with 2 e ( n ) many tails to create the wordsin W n +1 that belong to C . This puts 2 e ( n ) words into each C .Each equivalence class in Q n +1 n +1 consists of taking all words starting with a singlefixed stem. It is immediate that there are 2 e ( n ) many Q n +1 n +1 -classes in each Q n +1 n class and that each Q n +1 n +1 class has 2 e ( n ) many words in it. Moreover each class isassociated with a fixed stem of length k Max − k Max followed by many short tails.Thus specifications Q4 and Q6 are satisfied.Finally we note that W n +1 was built by n + 2 many successive substitutions ofsize 2 e ( n ) into equivalence classes. Thus s n +1 = 2 ( n +2) e ( n ) . Why does this work?
Though it appears in detail in [9], for the reader’s edifica-tion it may be appropriate to say a few things about how the stems/tails constructionaffects the statistics. This issue is most cogent in J10.1, where sh tK n ( u ) and v arebeing compared on small portions of their overlaps. By the manner of constructionof the stems, where the stem of sh tK n ( u ) overlaps with the stem of v conclusion 4of Proposition 20 holds with (cid:15) a = (cid:15) n / j ≥ (cid:15) n k n the total length of the overlap is at least (cid:15) n k n K n . The tails havelength k Max s n K n , so the proportion of the overlap taken up by the tails is at most2 k Max s n j ≤ k Max s n (cid:15) n k n < k Max (cid:15) n (cid:15) n < k Max (cid:15) n < (cid:15) n / . Cf. Propositions 66 and 65, and Section 8.3, in [9]. j < j where ( u (cid:48) , v (cid:48) ) occur.This proportion is the weighted average of the proportion P S of j < j where ( u (cid:48) , v (cid:48) )occur in the overlaps of the stems and the proportion P T of j < j where ( u (cid:48) , v (cid:48) )occur in an overlap of a stem with a tail. Let α be the proportion of the overlap of sh tK n ( u ) and v that occurs on the stems. By the above, α > − (cid:15) n /
50. Then (cid:12)(cid:12)(cid:12)(cid:12) r ( u (cid:48) , v (cid:48) ) j − s n (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ( αP S + (1 − α ) P T ) − s n (cid:12)(cid:12)(cid:12)(cid:12) On the overlap of the stems | P S − s n | < (cid:15) n / P T ∈ [0 ,
1] and (1 − α ) < (cid:15) n / (cid:12)(cid:12)(cid:12)(cid:12) r ( u (cid:48) , v (cid:48) ) j − s n (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) n . Hence J10.1 holds.
The action (cid:121) n +1 .Case 1 ( n + 1 ≥ Ω ): In this case, the action of (cid:121) n +1 is trivial, so there is nothingfurther to be done. Case 2 ( n + 1 < Ω ): In this case, we need to define (cid:121) n +1 to be subordinate to (cid:121) n . Fix a Q n +1 n -class C and suppose that C gets sent to D by (cid:121) n . Sinceeach Q n +1 n class has the same number of elements we can define (cid:121) n +1 so thatit induces a bijection between the Q n +1 subclasses of C and D . The construction of the W n , Q n and (cid:121) n is primitive recursive Here is astandard theorem:
Theorem 21 (Hoeffding’s Inequality) . Let (cid:104) X n : n ∈ N (cid:105) be a sequence of i.i.d.Bernoulli random variables with probability of success p . Then, P (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n n − (cid:88) k =0 X k − p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > δ (cid:33) < exp (cid:18) − nδ (cid:19) . Lemma 22.
The construction of the sequence (cid:104)W n , Q n , (cid:121) n : n ∈ N (cid:105) is primitiverecursive.Proof. The only part of the construction that is not a completely explicit induction isfinding the collection of words satisfying the conclusions of the Substitution Lemma.For each candidate fixed k one can primitively recursively search all substitutioninstances to see if there is a collection of words of length k ∗ K N that works. UsingHoeffding’s inequality can give an explicit upper bound for a k that works. Thealgorithm first computes a k n that works and then does the search. (cid:97) Remark 23.
Two remarks are in order. • The asymmetry of the words in the last step of the construction of W n +1 appears problematic. How can the words all be oriented left-to-right stem andtail if they are supposed to be closed under all the various skew-diagonal actionsat stage n + 1 and later?The answer is that the asymmetries are covered up by the equivalence classes.For example, the words in W n +1 / Q n +1 n +1 are all constant sequences of length K n +1 . If w ∈ W n +1 and C is the Q n +1 n +1 -class corresponding to w then the wordin W n +1 / Q n +1 n +1 corresponding to w is simply a string of K n +1 C ’s. Supposethat (cid:121) n +1 ( C ) = D . When the action (cid:121) n +1 is extended to the skew-diagonalaction at a later stage m , it simply takes this string of C ’s to a string of D ’s ina different place in a reverse word in the alphabet W n +1 / Q n +1 n +1 . It is completelyopaque whether the elements of D have tails on the same side or the oppositeside as the tails of words in C . • Roughly speaking, Cases 1 and 2 above correspond to Cases 1 and 2 in section8.3 of [9], albeit with several differences. A key one is that here, once theconstruction falls into Case 1, it remains in Case 1.We note that we have created inductive lower bounds on the size of k n . Numerical Requirement C k n is large enough that s n +1 ≤ s k n n . Numerical Requirement D k n is large enough to satisfy the use of the Substitu-tion Lemma 20 to construct the words in W n +1 . In particular /k n < (cid:15) n / .The data for numerical requirement D comes from the coefficients and words andequivalence relations at stages n − and before. By Theorem 9, we have a primitive recursive reduction F O from G¨odel numbers ofΠ sets to uniquely ergodic odometer-based systems. However the main theoremis about diffeomorphisms of the torus and it is an open problem whether there isany smooth ergodic transformation of a compact manifold that has an odometeras a factor. Rather than attack this problem directly, we follow [11] and do asecond transformation of odometer-based systems into circular systems , which canbe realized as diffeomorphisms. This is the downward vertical arrow F on the rightof figure 1.Subsection 3.1 covers circular systems and their construction. The primitiverecursive map F maps from the odometer-based systems to circular systems and32reserves synchronous and antisynchronous factors and conjugacies. In particular,for those odometer-based systems K in the range of F O , K is or is not isomorphicto its inverse, if and only if F ( K ) is or is not isomorphic to its inverse. We use thelanguage of category theory to describe the structure that is preserved and definethe categorical isomorphism.In Subsection 3.3, the circular systems produced are realized as smooth diffeo-morphisms of the torus. This is done in two steps: first, a given circular system isrealized as a discontinuous map of the torus; second, it is shown that how to smooththe toral map into a diffeomorphisms that is measure theoretically isomorphic tothe circular system. Like odometer-based systems, circular systems are symbolic systems characterizedby construction sequences (cid:104)W cn : n ∈ N (cid:105) of a certain form. The basic tool forconstructing circular systems is the C -operator. Let k, l, q ∈ N be arbitrary integers greater than 1, and p be coprime to q . Let0 ≤ j i < q indicate the unique integer such that j i · p = i (mod q ) . (12)We can rewrite j i as p − i (mod q ), and reserve the subscript notation for this use. Definition 24 (The C -Operator) . Let Σ be a non-empty finite alphabet and let b and e be two new symbols not contained in Σ . Let w , . . . , w k − be words in Σ ∪ { b, e } .The C -operator is given by: C ( w , . . . , w k − ) = k − (cid:89) i =0 q − (cid:89) j =0 b q − j i · w l − j · e j i , where “ (cid:81) ” indicates concatenation. Fix a sequence (cid:104) k n , l n : n ∈ N (cid:105) of positive integers with k n ≥ l n increasingand (cid:80) n /l n < ∞ . We follow Anosov-Katok ([1]) and define auxiliary sequences ofintegers (cid:104) p n : n ∈ N (cid:105) and (cid:104) q n : n ∈ N (cid:105) . Set q = 1, p = 0. Inductively define q n +1 = k n l n q n (13)and p n +1 = k n l n p n q n + 1 . (14)Note that p n and q n are coprime for n ≥ α n = p n /q n . Then α n +1 = α n + 1 /q n +1 . (15)Since q n > l n and (cid:80) n /l n < ∞ , we have (cid:80) n /q n < ∞ . Thus the α n converge toa Liouvillean irrational α ∈ [0 , α = lim n →∞ α n = (cid:88) n ≥ q n . (16) Circular Construction sequences
We first define the notion of a circular con-struction sequence. Fix a non-empty finite alphabet Σ ∪ { b, e } as above as well aspositive natural number sequences (cid:104) k n : n ∈ N (cid:105) and (cid:104) l n : n ∈ N (cid:105) , with k n ≥ (cid:104) l n (cid:105) strictly increasing such that (cid:80) ∞ n =1 /l n < ∞ . We take l = 1.Let W c = Σ. For every n , choose a set P n +1 ⊆ ( W cn ) k n of prewords . Then W cn +1 is given by all words of the form C ( w , . . . , w k n − ) = k n − (cid:89) i =0 q n − (cid:89) j =0 b q n − j i · w l n − j · e j i (17)where ( w , . . . , w k n − ) ∈ P n +1 is a preword. We call C the C -operator .The words created by the C -operator are necessarily uniquely readable. However,we further demand that the collections of prewords (cid:104) P n : n ∈ N (cid:105) are uniquelyreadable in the sense that each k n -tuple of words p ∈ P n +1 , considered a word in thealphabet W cn , is uniquely readable. (Unique readability is discussed in AppendixC.3 in definition 49. See the discussion in [13] for more details. ) Definition 25 (Circular system) . Let (cid:104)W cn : n ∈ N (cid:105) be a circular constructionsequence. Then the limit, which we denote K c is a circular system . To emphasize that a given construction sequence is circular we denote it (cid:104)W cn : n ∈ N (cid:105) . In this paper the circular construction sequences will be strongly uniform. Asa consequence the resulting symbolic shift is uniquely ergodic and we can write K c = ((Σ ∪ { b, e } ) Z , B , µ, Sh ) where µ is the unique shift invariant measure on K c . Example 26.
Let
Σ = {∗} . Then |W c | = 1 . Passing from from W cn to W cn +1 oneinductively shows that for all n, |W cn | = 1 . Define K α to be the limit of the resultingconstruction sequence.Suppose that (cid:104)U cn : n ∈ N (cid:105) is another circular construction sequence in an alpha-bet Λ with the same coefficients (cid:104) k n , l n : n ∈ N (cid:105) having a limit L c . Define a map : L c → K α by setting π ( f )( n ) = ∗ if f ( n ) ∈ Λ b if f ( n ) = b,e if f ( n ) = e. Then π is a factor map of symbolic systems. Hence K α is a factor of every circularsystem with coefficients (cid:104) k n , l n : n ∈ N (cid:105) . For α ∈ [0 , R α : S → S be rotation by 2 πα radians. Equivalently we view R α : [0 , → [0 ,
1) as given by x (cid:55)→ x + α (mod 1). This rotation R α plays thesame role with respect to circular systems as the canonical odometer factor playswith respect to the odometer-based systems of Section 2. Lemma 27 (The Rotation Factor) . Let α = lim α n be defined from a sequence (cid:104) k n , l n : n ∈ N (cid:105) from equation 16. Then K α ∼ = R α . In particular if ( K c , B , ν, Sh ) isa circular system in the alphabet Σ ∪ { b, e } with parameters (cid:104) k n , l n : n ∈ N (cid:105) , thenthere is a canonical factor map ρ : K c → R α .Proof sketch. For almost every x ∈ K α , there is an N for all n ≥ N there are a n , b n ≥ x (cid:22) [ − a n , b n ) is some word in W cn . All words in W cn have thesame length, q n , so we can define the following quantity: ρ n ( x ) = a n (cid:18) p n q n (cid:19) . Straightforward algebraic manipulations give that (cid:12)(cid:12)(cid:12)(cid:12) ρ n +1 ( x ) − ρ n ( x ) < q n (cid:12)(cid:12)(cid:12)(cid:12) whence it is clear that ρ n ( x ) → ρ ( x ) ∈ [0 , ρ n ( Sh ( x )) = ρ n ( x ) + p n q n taking limits shows that ρ ( Sh ( x )) = ρ ( x ) + α , as desired. (cid:97) See Theorem 52 in [11] for a complete proof.35 istinguishing α ’s Theorem 1 demands that if M (cid:54) = N , then F ( M ) (cid:54)∼ = F ( N ).This is achieved by arranging that the Kronecker factors of F ( M ) and F ( N ) arenon-isomorphic rotations of the circle. This requires that α ( N ) (cid:54) = α ( M ) and that K α ( N ) is the Kronecker factor of the limit sequence K c ( N ). Recall that for each N we have a prime number P N which we take for k and and we build sequences (cid:104) k n ( N ) , l n ( N ) : n ∈ N (cid:105) , which in turn, yield sequences (cid:104) p n , q n , k n , l n : n ∈ N (cid:105) ( N )and (cid:104) α n ( N ) : n ∈ N (cid:105) which converge to an irrational α ( N ).For each N we take l ( N ) = 1, so α ( N ) = P N . The sequence (cid:104) k n ( N ) : n ∈ N (cid:105) is defined in the construction of the odometer construction sequences as describedafter Lemma 20. The l n ’s are chosen in the construction of the circular sequencesand diffeomorphisms. They must satisfy some lower bounds on their growth, whichwe describe later.To ensure different rotation factors correspond to different Π sentences, we alsoput the following growth requirement on the (cid:104) l n ( N ) : n ∈ N (cid:105) sequences: Numerical Requirement E
Growth Requirement on the l n ’s: l n ( N ) ≥ l n ( N − . Lemma 28.
Suppose that the k n ( N − , k n ( N ) , l n ( N − and l n ( N ) satisfy Re-quirements B and E. Then α ( N − > α ( N ) . (cid:96) Note that k ( N −
1) = P N − < P N = k ( N ), so q ( N − < q ( N ) Since q n +1 = k n l n q n , k n ( N ) ≥ k n ( N −
1) and l n ( N ) ≥ l n ( N −
1) one sees inductively thatfor all n q n ( N − ≤ q n ( N ).By equation 16 we see that α ( N −
1) = (cid:88) n ≥ q n ( N − > (cid:88) n ≥ q n ( N ) = α ( N ) . (cid:97) Synchronous and Anti-synchronous joinings
The system K α gives a symbolicrepresentation of the rotation R α by 2 πα radians. The inverse transform Rev ( K α ) istherefore a representation of rotation by 2 π (1 − α ) ≡ π ( − α ) radians. Moreover theconjugacies ϕ : S → S between R α and R − α = R − α are of the form z (cid:55)→ ¯ z ∗ e πiδ forsome δ . For combinatorial reasons we fix a particular conjugacy (cid:92) : K α → Rev ( K α )that is described explicitly in [12]. Thus (cid:92) is given by the map defined on S by z (cid:55)→ ¯ z ∗ e πiγ for some particular γ . In additive notation on [0 ,
1) this becomes x (cid:55)→ − x + δ ( mod 1) for some δ . 36he importance of rotation factors and odometer factors in the sequel is theirfunction as “timing mechanisms.” Joinings between odometer-based systems inducejoinings on the underlying odometers; the same holds true of circular systems. Definition 29 (Synchronous and Anti-synchronous Joinings) . We define two kindsof joinings, synchronous and anti-synchronous . • Let K and K be odometer-based systems sharing the same parameter sequence (cid:104) k n : n ∈ N (cid:105) . Let η be a joining between K and K . Then η induces a joining η π between K and K ’s copies of the underlying odometer O . The joining η is synchronous if η π is the graph joining corresponding to the identity map from O to O . A joining η between K and K is anti-synchronous if η π is the graphjoining corresponding to the map x (cid:55)→ − x from O to O − . • Let K c and K c be circular systems sharing the same parameter sequence (cid:104) k n , l n : n ∈ N (cid:105) . Let η be a joining between K c and K c . Then η induces a joining η π between K c and K c ’s copies of the rotation factor, K α . The joining η is syn-chronous if η π is the graph joining corresponding to the identity on K α × K α .A joining η between K c and ( K c ) − is antisynchronous if η π restricts to thegraph joining corresponding to (cid:92) : K α → ( K α ) − . Odometer-based systems and Circular systems that share the same parameter se-quence (cid:104) k n : n ∈ N (cid:105) have similar joining structures. We begin by defining twocategories.Fix a parameter sequences (cid:104) k n : n ∈ N (cid:105) and (cid:104) l n : n ∈ N (cid:105) with (cid:80) /l n < ∞ . Let O B be the category whose objects consist of all ergodic odometer-based systemswith coefficients (cid:104) k n : n ∈ N (cid:105) . A morphism of O B is either a synchronous graphjoining between K and L or an anti-synchronous graph joining between K and L − .Let C B be the category whose objects consist of ergodic circular systems built withcoefficients (cid:104) k n , l n : n ∈ N (cid:105) and whose morphisms consist of synchronous and anti-synchronous graph joinings from K c with ( L c ) ± .The main result of [12] is the following: Theorem 30 (Global Structure Theorem) . The categories O B and C B are iso-morphic by a functor F that takes synchronous joinings to synchronous joinings,anti-synchronous joinings to anti-synchronous joinings and isomorphisms to iso-morphisms. To prove Theorem 30 one must define the map F on objects, and on morphismsand then show that it is a bijection and preserves composition. Since we will only beconcerned here with how effective F is we confine ourselves to defining it and referthe reader to [12] for complete proofs. In [14], the proof is discussed to understandthe strength of the assumptions needed to prove it.We begin by defining F on the objects.37 efining F on objects. Let an K be an odometer-based system with associatedconstruction and parameter sequences (cid:104)W n : n ∈ N (cid:105) and (cid:104) k n : n ∈ N (cid:105) . Let (cid:104) l n : n ∈ N (cid:105) be an arbitrary sequence of positive integers growing fast enough that (cid:80) n /l n < ∞ . Inductively define a map F taking the construction sequence for anodometer-based system K to a construction sequence for a circular system K c byapplying the C -operator. Define maps c n : W n → W cn as follows: • Let W c = Σ and c be the identity. • Suppose that c n and W cn have been defined. Let W cn +1 = {C ( c n ( w ) , . . . , c n ( w k n − )) : w w · · · w k n − ∈ W n +1 } and w i ∈ W n . Define c n +1 : W n +1 → W cn +1 by setting c n +1 ( w · · · w k n − ) = C ( c n ( w ) , . . . , c n ( w k n − )) . where w i ∈ W n with w · · · w k n − ∈ W n +1 .The construction sequence (cid:104)W cn : n ∈ N (cid:105) then gives rise to a circular system K c .The functor F will associate K c with K . Lifting measures and joinings
We need to lift measures on odometer basedsystems to measures on circular systems for two reasons:1. To complete the definition of F on objects, given an odometer based system( K , µ ) we need to canonically associate a measure µ c to K c . Then F ( K , µ ) =( K c , µ c ).In the context of this paper this first reason is not pressing: the constructionsequences in the range of F O are strongly uniform, hence uniquely ergodic.Thus there is only one candidate for µ c . However to complete the definitionof F we need to understand what happens for arbitrary ergodic µ .2. To define F on morphisms, given a joining J between ( K , µ ) and ( L , ν ) weneed to associate a joining J c between ( K c , µ c ) with ( L c , ν c ).For the second issue, and to deal with general odometer based systems ( K , µ ),we review the notion of generic sequences of words. These were introduced in[24] and used in the proof of Theorem 30 [12].Let k, l > (cid:104)W n : n ∈ N (cid:105) be an arbitrary construction sequence. Using theunique readability of words in W k a word w in Σ q k + l determines a unique sequenceof words w j in W k such that , w = u w u w . . . w J u J +1 . w ∈ W k + l , each u j is in the region of spacers added in W k + l (cid:48) , for l (cid:48) ≤ l . Wewill denote the empirical distribution of W k -words in w by EmpDist k ( w ). Formally:EmpDist k ( w )( w (cid:48) ) = |{ ≤ j ≤ J : w j = w (cid:48) }| J + 1 , w (cid:48) ∈ W k . Then
EmpDist extends to a measure on P ( W k ) in the obvious way.To finitize the idea of a generic point for a system ( K , µ ) we introduce the notionof a generic sequence of words. By µ m we will denote the discrete measure on thefinite set Σ m given by µ m ( u ) = µ ( (cid:104) u (cid:105) ). Then µ m is not a probability measure so wenormalize it. Let ˆ µ n ( w ) denote the discrete probability measure on W n defined byˆ µ n ( w ) = µ q n ( (cid:104) w (cid:105) ) (cid:80) w (cid:48) ∈W n µ q n ( (cid:104) w (cid:48) (cid:105) ) . Thus ˆ µ n ( w ) is the relative measure of (cid:104) w (cid:105) among all (cid:104) w (cid:48) (cid:105) , w (cid:48) ∈ W n . The de-nominator is a normalizing constant to account for spacers at stages m > n and forshifts of size less than q n . Definition 31.
A sequence (cid:104) v n ∈ W n : n ∈ N (cid:105) is a generic sequence of words if andonly if for all k and (cid:15) > there is an N for all m, n > N , (cid:107) EmpDist k ( v m ) − EmpDist k ( v n ) (cid:107) var < (cid:15). The sequence is generic for a measure µ if for all k : lim n →∞ (cid:107) EmpDist k ( v n ) − ˆ µ k (cid:107) var = 0 where (cid:107) (cid:107) var is the variation norm on probability distributions. The point here is that the ergodic theorem gives infinite generic sequences formeasures µ . These infinite generic sequences in turn, create generic sequences offinite words. A generic sequence of finite words determines a measure. If the genericsequence is built from the measure then the measure it determines is the originalmeasureWe now deal with the first issue above for arbitrary ( K , ν ) (and not just thosethat are strongly uniform). Given an odometer based system ( K , ν ) we must specifythe measure ν c we associate with ν . Section 2.6 of [12] gives a canonical methodof constructing a generic sequence of words (cid:104) v n : n ∈ N (cid:105) that encode any ergodicmeasure on K . The corresponding sequence of words v cn = c n ( v n ) is also generic anddetermines an ergodic measure on K c . The map F then takes ( K , ν ) to ( K c , ν c )39 efining F on morphisms Given an arbitrary synchronous or anti-synchronousjoining J between odometer based systems K and L ± we can view ( K × L , J ) asan odometer based system. Taking a generic sequence of pairs of words (cid:104) ( u n , v n ) : n ∈ N (cid:105) for J as in [12] and lifting it with the sequence of c n ’s (and adjustingappropriately for reversing the circular operation with a mechanism denoted (cid:92) in[12]), one gets a joining J c between K c and L c .Define F ( J ) = J c . Is F primitive recursive? Clearly the maps c n are primitive recursive so themap taking a construction sequence (cid:104)W n : n ∈ N (cid:105) to (cid:104)W cn : n ∈ N (cid:105) is primitiverecursive. For the same reason the map taking a joining J specified by a givengeneric sequence to J c is primitive recursive. Thus, assuming that joinings J arepresented in a manner that one can compute the generic sequences of words, themap J (cid:55)→ J c is primitive recursive.In the context of the systems in the range of F O , the relevant joinings between K and K − are given by limits of η n ’s, and the generic word sequences are easilyseen to be primitive recursive and can thus be translated to the joinings of K c with( K c ) − . Remark 32.
We have shown that if ϕ N is true then F ◦ F O ( N ) is isomorphic to F ◦ F O ( N ) − and the isomorphism is primitive recursive. In section 3.3 we build aprimitive recursive realization function R which maps from strongly uniform circularsystems to measure preserving diffeomorphisms of the torus. Since F = R ◦ F ◦ F O ,the result we prove is something stronger than claimed in Theorem 1. Namely weshow that if ϕ N is true then there is a measure isomorphism between F ( N ) and F ( N ) − coded by a primitive recursive generic sequence of words. The second clause of the Main Theorem (Theorem 1) says that if M and N aredistinct natural numbers than the corresponding diffeomorphisms T M and T N arenot isomorphic. To distinguish between them we use their Kronecker factors. (Formore information on the Kronecker factors, see e.g. [23].) For this purpose weprove the following proposition. This section is otherwise independent of the othersections. Readers who find the proposition and corollary obvious can skip to thenext section. Proposition 33.
Let K c be circular system in the range of F ◦ F O , built withcoefficients (cid:104) k n , l n : n ∈ N (cid:105) and α = lim n α n then the Kronecker factor of K c ismeasure theoretically isomorphic to the rotation R α . An immediate corollary of this is: See section 2.3 for an explanation of the ( N )-notation. orollary 34. Suppose that
M < N are natural numbers. Then:1. α ( N ) < α ( M ) , where α ( N ) and α ( M ) are the irrationals associated with therotation factors of F ( N ) and F ( M ) .2. ( K c ) M (cid:54)∼ = ( K c ) N . (cid:96) This follows immediately from Lemma 28 and the fact that the Kronecker factor( K c ) M is isomorphic to R α ( M ) and the Kronecker factor of ( K c ) N is isomorphic to R α ( N ) . (cid:97) After Proposition 33 is shown we will have proved the following intermediatestep in the proof of Theorem 1:
Proposition 35.
For N a code of a Π sentence, then F ◦ F O ( N ) is a primitiverecursive circular construction sequence and1. N is the code for a true statement if and only if the circular system T deter-mined by F ◦ F O ( N ) is measure theoretically conjugate to T − ;2. F ◦ F O ( N ) is ergodic–in fact strongly uniform; and3. For M (cid:54) = N , F ◦ F O ( M ) is not conjugate to F ◦ F O ( N ) . Review of the Kronecker factor
Let (cid:126)γ = (cid:104) γ m : m ∈ Z (cid:105) be an enumeration ofthe eigenvalues of the Koopman operator of a measure preserving transformation( X, B , µ, T ). Then (cid:126)γ determines a measure preserving action on (( S ) Z , λ Z ) (where λ Z is the product measure on ( S ) Z ) by coordinatewise multiplication. The actionis ergodic, discrete spectrum and isomorphic to the Kronecker factor of ( X, B , µ, T ).If α is an eigenvalue of the shift operator then the powers of α , (cid:126)α = (cid:104) α n : n ∈ Z (cid:105) are also eigenvalues corresponding to a subsequence of (cid:126)γ and hence thecoordinatewise multiplication of (cid:126)α on ( S ) Z determines a factor of the Kroneckerfactor. This is a proper factor if and only if there is an eigenvalue of the Koopmanoperator that is not a power of α . In particular there is a non-trivial projection mapfrom the Kronecker factor to the dual of the countable group { α n : n ∈ Z } The proof of Proposition 33 follows the outline of the proof of Corollary 33of [9]. Working in the context of odometer based systems built with coefficients (cid:104) k n : n ∈ N (cid:105) , it says that the Kronecker factor K r of each system K in the rangeof F O is the odometer transformation O based on (cid:104) k n : n ∈ N (cid:105) . Note that theodometer O is a subgroup of the Kronecker factor since rotation by the k thn root ofunity is an eigenvalue of the Koopman operator. The steps there are:1. Any joining J of K with K projects to a joining J O of O with itself. If J O isnot given by the graph joining coming from a finite shift of the odometer then J must be the relatively independent joining of K with itself over J O . (Thisis Proposition 32 of [9].) 41. If there is an eigenvalue of the unitary operator associated with K that is nota power of α there is a non-identity element t in the Kronecker factor whoseprojection to the odometer O is the identity.3. Multiplying t by an element h ∈ O which is not a finite shift gives an element t (cid:48) of the Kronecker factor K r that is not in O and projects to an element of O that is not a finite shift.4. Let H ∗ be the sub- σ -algebra of the measurable subsets of K generated by K r . Then multiplication by t (cid:48) gives a graph joining J ∗ of H ∗ with itself thatprojects to the joining of O given by multiplication by h . Extend J ∗ to ajoining J of K with K . Then J does not project to a finite shift of theodometer but it is also not the relatively independent joining of K with itselfover the joining of O with itself given by h . This is a contradiction.To imitate this argument we first note that for circular systems, the analogueof the odometer is the rotation K α , and that every element β ∈ S determines aninvertible graph joining S β of K α with itself, corresponding to multiplication by β inthe group S . We need to identify the analogue of the “finite shifts on the odometer”in the case of circular systems. The appropriate notion is given in Definition 78 in[11], namely the central values . The central values form a subgroup of the unitcircle.To prove Proposition 33, fix a circular system K c in the range of F ◦ F O . Wefirst show that there is a β ∈ S that is not a central value. This β plays the rollof h in the outline given above. Then the analogue of Proposition 32 is proved: anyjoining of K c with itself that does not project to the joining given by multiplicationon S by a central value is the relatively independent joining over its projection.Suppose now that there is an eigenvalue of the Koopman operator that is not apower of α . Then the action of (cid:126)α on ( S ) Z is a non-trivial projection of the Kroneckerfactor K r c of K c . Hence we can fix a non-identity element t of the Kronecker factorwhose projection to the factor determined by the powers of α is the identity. As instep 3 above we multiply t by a non-central β to get a t (cid:48) in the Kronecker factorwhich:a.) induces a joining J ∗ of H ∗ with itself that projects to the graph joining of K α with itself induced by S β .Extending J ∗ to a joining J of K c with itself we see that:b.) J is not the relatively independent joining over the joining of K α given by S β .After the details are filled in, this contradiction establishes Proposition 33. Notation
As in previous sections we identify the unit interval [0 ,
1) with the unitcircle via the map x (cid:55)→ e πi ∗ x , which identifies “addition mod one” on the unit42nterval with multiplication on the unit circle. When we write “+” in this sectionit means addition mod one, interpreted in this manner.We use the following numerical requirement in explicit proof of Proposition 33: Numerical Requirement F
The k n ’s must grow fast enough that (cid:80) n k n < ∞ .To finish the proof of Proposition 33, we fix a circular system K c in the range of F ◦ F O and prove the following two lemmas. Lemma 36.
There is a non-central value β . Lemma 37.
Suppose that β is not a central value. Let J be a joining of K c × K c whose projection to K α × K α is the graph joining of K α with itself given bymultiplication by S β . Then J is the relatively independent joining of K c × K c overthe joining of K α with itself given by S β . (cid:96) [Lemma 36] While it seems very likely that there is a measure one set of exampleswe just need one. The example will be of the form β = (cid:80) ∞ n =1 a n k n q n for an inductivelychosen sequence of natural numbers (cid:104) a n : n ∈ N (cid:105) with 0 ≤ a n < n .To describe β completely and verify it is non-central we need several facts fromsections 5, 6 and 7 in [11], which discuss the relationship between the geometric andthe symbolic representations of K α .The geometric construction builds a sequence of periodic approximations oflengths (cid:104) q n : n ∈ N (cid:105) with the resulting limit being the rotation of the circle by R α .Expanding on Lemma 27, these approximations are given by the towers of intervals T n = { [0 , /q n ) , [ p n /q n , p n /q n + 1 /q n ) , [2 p n /q n , p n /q n + 1 /q n ) , . . . [ kp n /q n , kp n /q n +1 /q n ) , . . . } viewed as a periodic system.The symbolic representation uses the C operation to build the construction se-quence for the symbolic system. The latter is described in Example 26. We give thegeometric description of the periodic approximations presently.We use the following notions and notation from [11]:1. ϕ : ( K α , sh ) → ([0 , , +) is the measure theoretic isomorphism between theshift on K α and the rotation R α given in Lemma 27. We use s ’s to refer toelements of K α and x ’s to refer to elements of [0 ,
1) and s corresponds to x if ϕ ( s ) = x .2. The notion of s ∈ K α being mature implies that s has a principal n -subwordand it is repeated multiple times both before and after s (0).3. S β = ϕ − R β ϕ is the symbolic conjugate of the rotation R β , via the map ϕ . If s corresponds to x then S β ( s ) corresponds to x + β (mod 1). We willoccasionally be sloppy and use the language s + β for s ∈ K α when we mean S β ( s ). 43. In [11], a set S is defined as the collection of elements s ∈ K α such that theleft and right endpoints of the principal n -subwords of s go to minus and plusinfinity respectively. Explicitly suppose that s ∈ K α is such that for all largeenough n , the principal n -subword exists and lives on an interval [ − a n , b n ] ⊆ Z .The point s ∈ S if lim n a n = lim n b n = ∞ .The set S β is (cid:84) n ∈ Z S β ( S ), the maximal S β invariant subset of S . It is ofmeasure one for the unique invariant measure on K α . Since K α is a factor ofevery circular system K c with the same coefficient sequence (cid:104) k n , l n : n ∈ N (cid:105) forall invariant measures µ on K c , { s ∈ K c : the left and right endpoints of theprincipal n -subwords of s go to minus and plus infinity is of µ -measure one.5. Given an arbitrary β we can intersect S β with S q for all rational q and getanother set of measure one. Hence in a slight abuse of notation we assumethat S β is invariant under conjugation by rational rotations.6. For s ∈ K α , if r n ( s ) = i and x is the corresponding element of [0 ,
1) then x is in the i th level of the tower corresponding to the n th approximation to R α .This tower is given by R α n .In the geometric picture, at stage n , we have a tower of intervals of the form[ iα n , iα n + 1 /q n ) ordered in the dynamical ordering–where the successor of the in-terval [ iα n , iα n + 1 /q n ) is [( i + 1) α n , ( i + 1) α n + 1 /q n ). Thus level I i +1 is I i + α n .Passing from stage n to stage n + 1 involves subdividing the old levels into newlevels, which are of the form [ iα n +1 , iα n +1 + 1 /q n +1 ). These subintervals move diag-onally up and to the right through the n -levels. The diagonal movement correspondsto addition of α n +1 . The key formula is 15: α n +1 = α n + 1 /q n +1 . As illustrated in diagrams 9 and 10 of [13] and Figure 3, the n + 1 tower proceedsdiagonally up through the n -tower. This is evident from the form of equation 15and the fact that q n +1 = k n l n q n . 44 i g u r e : T h e n - t o w e r , s h o w i n g t h e d i ago n a l p r og r e ss i o n o f t h e n + - t o w e r . T h e h e a vy h o r i z o n t a lli n e s a r e t h e l e v e l s ( i α n , i α n + / q n ) , s t a r t i n g w i t h [ , / q n ) o n t h e b o tt o m . T h e l e v e l s t h e n + - t o w e r a r e t h e h o r i z o n t a l s e g m e n t s b e t w ee n t h e d i ago n a lli n e s . j/k n q n , ( j + 1) /k n q n ) sub-divisions of the levels. Those diagonals correspond to the boundary portion of the n + 1-words (the b ’s and the e ’s). The diagonal paths that start in the region[ j/k n q n , ( j + 1) /k n q n ) and traverse from the bottom to the top level while stayingin that region correspond to the j th argument of the C operator at stage n .Restating this in terms of the isomorphism ϕ between K α and ([0 , , R α ), if s and s are mature elements of K α corresponding to x , x ∈ S , then: • x , x belong to two diagonal strips that do not touch a vertical strip and withbase in the same interval [ j/k n q n , ( j + 1) /k n q n )iff • Inside their principal n + 1-subwords, s (0) and s (0) are in n -words comingfrom the same argument w αj of C ( w α , w α , . . . w αk n − ).We now continue the enumeration of basic notions in [11] we use here.7. In a very slight variation of the notation of [11], when we are comparing s with t we define d n ( s, t ) = r n ( t ) − r n ( s ) (mod q n ). In this argument frequently t = S β ( s ) and if β is clear from the context we simply write d n ( s ). If x and y correspond to s and t , the number d n ( s, t ) can be viewed either as thenumber of levels in the n -tower between x and y or as the difference betweenthe locations of 0 in the principal n -subwords of s and t .8. For mature s and t , the result of shifting t by − d n ( s, t ) units is that the locationof 0 is in the same position in its principal n -subword as is the position of s (0)in its principal n -subword.9. Applying the shift map d n +1 ( s, t ) times to s moves its zero to the same pointas t ’s is relative to its n + 1 subword. Subsequently moving it back − d n ( s, t )steps moves the zero of result back to the same position in its n -subword aszero is in s ’s n -subword. In other words, if s (cid:48) is the result of applying the shiftmap to s d n +1 ( s, t ) − d n ( s, t ) times, then 0 is in the same position relative tothe n -block of s (cid:48) as it is in s .10. The n + 1-word in the construction sequence for K α is of the form C ( w α , w α . . . w αk n − )and hence if s is mature at stage n then s (0) occurs in an n -block correspondingto the position of w αj for some j . We can ask whether the j correspondingto the principal n -subword of the ( d n +1 ( s, t ) − d n ( s, t ))-shift of s is the sameas the j corresponding to the principal n -subword of s .46f it does, then s and t are well-matched at stage n and if not the s and t are ill-matched at stage n .11. If s (cid:48) is the result of shifting s d n +1 ( s, t ) − d n ( s, t ) times then the 0 of s (cid:48) is inthe same w αj as is the zero of t . So for the purposes of determining whether s is well- β -matched at stage n , we can compare which argument of C s (0) and t (0) belong to. As a result we can speak of s and t well or ill-matched at stage n . If x and y are the corresponding members of [0 ,
1) we can say say that x and y are well or ill-matched at stage n .We are now ready to construct the non-central β . We do this by induction. Atstage 1, a = 0. At stage n , we let β n = (cid:80) n − p =1 a p k p q p . For i = 0 , . . . n −
1, in theterminology of item 11 consider M i = { x : x and x + β n + i/k n q n are well-matched } . Since the M i ’s are disjoint, for some i , λ ( M i ) ≤ n . Let a n be such an i and let β n +1 = β n + a n /k n q n . Finally we let β = (cid:80) ∞ a p k p q p , so β = lim n →∞ β n .To see this works, we first show that:For almost all x , for large enough m , if s m corresponds to x + β m thenfor all mature n ≤ m, r n ( s m ) = r n ( S β ( s )).This is a Borel-Cantelli argument. Note that if r m ( s m ) = r m ( S β ( s )) then for allmature n ≤ m, r n ( s m ) = r n ( S β ( s )). Hence it suffices to show that for almost all s ,all sufficiently large m , r m ( s m ) = r m ( S β ( s )).If x corresponds to s then the only way that r m ( s m ) (cid:54) = r m ( S β ( s )) is if x + β m isin a different level of the m -tower than x + β m + (cid:80) ∞ p = m a p k p q p . In turn, the only waythat this can happen is if for some i , x + β m ∈ [ iα m + 1 /q m − ∞ (cid:88) p = m a p k p q p , iα m + 1 /q m ) . The latter interval is the right hand portion of a level in the m -tower, i.e. of aninterval of the form [ iα m , iα m + 1 /q m ).The collection of x that have this property for a given level i has measure (cid:80) ∞ p = m a p k p q p . Since there are q m many levels i , the measure of all of the x withthis property at stage m is q m ∗ (cid:16)(cid:80) ∞ p = m a p k p q p (cid:17) . Computing: q m ∗ (cid:32) ∞ (cid:88) p = m a p k p q p (cid:33) = a m k m + q m ∗ ∞ (cid:88) m +1 a p k p q p < a m k m + q m q m +1 ∞ (cid:88) m +1 a p k p ≤ a m k m + 1 k m l m q m C, C = (cid:80) ∞ a p k p . Since we assume that (cid:80) n k n < ∞ and a p < n − , C is finite. Wesee immediately that the measures of the collections of x such that at some stage m the level of x + β m in the m -tower is different from the level of x + β in the m -toweris summable. By Borel-Cantelli, it follows that for almost all s there is an N for all m ≥ N , r m ( s m ) = r m ( S β ( s )).From the choice of a n for all but a set of measure at most 1 / n , the s are ill-matched with s n +1 . Again by the Borel-Cantelli lemma, for almost all s there is an N for all n ≥ N s and s n +1 are ill-matched. Since for almost all s and all largeenough n the level of s n +1 is equal to the level S β ( s ) it follows that for almost all s and all large enough n s is ill-matched with S β ( s ). If ν is the unique invariantmeasure on K α then equation 33 of [11] defines∆ n ( β ) = ν ( { s : s is ill- β -matched at stage n } ) . We have shown that ∆ n ( β ) → n
1. Hence∆( β ) = (cid:88) n ∆ n ( β )is infinite. Hence we have shown that β is not central. (cid:97) We now prove Lemma 37 (cid:96) [Lemma 37] First note that the analysis in section 6.3, on page 50 of [11], saysthat for any non-central β we can choose hd and hd and a spaced out set G suchthat, as in equation 35 on page 50, letting (cid:54)⇓ n = { s : s is ill- β -matched at stage n and in configuration P hd ,hd } we get equation 36 on page 50 of [11]: (cid:88) n ∈ G ν ( (cid:54)⇓ n ) = ∞ . (18)We now observe that for n < m ∈ G , (cid:54)⇓ n and (cid:54)⇓ m are probabilistically indepen-dent. This follows from Lemma 75 on page 47 of [11]: belonging to (cid:54)⇓ n is an issue ofthe value of d n +1 − d n . The differences d m +1 − d m are independent of the differences d n +1 − d n , hence the sets (cid:54)⇓ n and (cid:54)⇓ m are pairwise independent. Which level x is onin the n -tower is independent of whether or not x is misaligned at the next stage.Let M m be the collection of s that are mature at stage m . Then applyingthe “hard” Borel-Cantelli lemma, for almost all s ∈ M m , there are infinitely many n ∈ G, s ∈(cid:54)⇓ n . Since (cid:83) m M m has measure one, for almost all s ∈ K c there areinfinitely many n, s ∈(cid:54)⇓ n .We now argue that if J and J are two joinings of K c × K c over S β , then theyare identical. Thus they are both the relatively independent joining. The resultfollows from the following claim which is an analogue of the claim in Proposition 32of [9]: 48 laim Let J be a joining of K c with itself that projects to the graph joining of K α with itself given by S β . Then for all cylinder sets (cid:104) a (cid:105) × (cid:104) b (cid:105) in K c × K c , the densityof occurrences of ( a, b ) in a generic pair ( x, y ) for J does not depend on the choiceof ( x, y ). (cid:96) Since β is non-central, and x, y are generic and J extends S β , we know that forinfinitely many n ∈ G the n -words of x and y are misaligned. Let G ∗ be this set.It suffices to show that: • There is a sequence of subblocks of the principal n + 1-subwords of x and y oftotal length B n , • as n ∈ G ∗ goes to infinity, B n /q n +1 goes to 0, • after removing the subwords in B n the number of occurrences of (cid:104) a (cid:105) × (cid:104) b (cid:105) isindependent of the choice of ( x, y ).Fix a large n ∈ G ∗ . We count occurrences of ( a, b ) in ( x, y ) over the portion ofthe principal n + 1-subwords of x that overlap with the n + 1-blocks of y . As inProposition 32 of [9], we show that, up to a negligible portion, this is independentof ( x, y ). From the definition of (cid:54)⇓ n for n ∈ G , there are fixed values of hd and hd .The number hd determines the overlap of the n + 1-block of x containing x (0) isthe left or right overlap. For convenience, assume that hd = L and hd = R .First: discard n -subwords that are not mature. This is a negligible portion.Next, shift y back by d n ( x ), so that the mature n -subwords of x in the principal n + 1-subword are aligned along the overlap of the principal n + 1 subword of y withthe corresponding n -subword of y . Then by specification J.10.1 and the fact that x and y are misaligned, any pairof n –words ( u, v ) occurs almost exactly 1 /s n times. So, after discarding a negligibleportion of the occurrences all pairs occur the same number of times. Shifting themall back by d n ( x ), an amount determined by β and thus independent of x and y ,gives a collection of counts of occurrences of ( a, b ) in all pairs ( u, sh d n ( v )) with allpairs occurring essentially the same number of times. The result is independent ofthe choice of x and y . The errors from the negligible portions and they go to zeroin proportion to n + 1. This proves the claim. (cid:97) The map
F ◦ F O maps codes for Π sentences to construction sequence for circu-lar systems. We now indicate how to realize circular systems as diffeomorphismsand why these diffeomorphisms are computable. The realization map is describedcompletely in [13]. We review it here to verify its effectiveness. Sections 4.3-4.6 of [11] discuss how the spacings of left and right overlaps correspond. , × [0 ,
1] with appropriateedges identified, is divided into rectangles. These are then permuted by the periodictransformations according to the action of the shift operator on the circular system.In the first stage, this permutation is built without regard to continuity. The resultis an abstract measure preserving transformation. In the second part, using smoothapproximations to these permutations, the limit is a C ∞ diffeomorphism.The main tool for moving from the discontinuous, symbolic transformationsto the smooth geometric transformations is the Anosov-Katok method of Approx-imation by Conjugacy [1]. To allow for this smoothing the parameter sequence (cid:104) k n , l n : n ∈ N (cid:105) must have the sequence of l n ’s grow sufficiently fast.The lower bounds l ∗ n ( (cid:104) k m : m ≤ n (cid:105) , (cid:104) l m : m < n (cid:105) ) will be determined inductively,the complete list of requirements on l ∗ n appears in Appendix A.For the moment we assume we are given the circular sequence (cid:104)W cn : n ∈ N (cid:105) with prescribed coefficient sequences (cid:104) k n , l n : n ∈ N (cid:105) where the l n grow sufficientlyfast.The periodic approximations to the first stage transformation are of the form T n = Z n ◦ R α n ◦ Z − n (19)which result from conjugating horizontal rotations ( x, y ) (cid:55)→ R αn ( x + α n , y ), withthe more complicated transformations h n : T → T that permute rectangularsubsets of T . The α n are the rationals constructed from the coefficient sequence (cid:104) k n , l n : n ∈ N (cid:105) described in section 3.1.1. The maps Z n are of the form Z n = h ◦ h ◦ . . . ◦ h n where h i codes the combinatorial behavior of the i th application of the C -operation.The initial, discontinuous transformation T will then be the almost-everywherepointwise limit of the sequence (cid:104) T n : n ∈ N (cid:105) .In the second part of the construction the h n ’s will be replaced by smooth trans-formations h sn that are close measure theoretic approximations to the h n ’s. Thisresults in a new sequence H n = h s ◦ h s ◦ · · · ◦ h sn . (20)The analogue of equation 19 for the final smooth transformation is: S n = H n R α n H − n (21)The sequence of S n ’s converge in the C ∞ -topology to a C ∞ measure preservingtransformation S : T → T . 50 hy do we do this? In [13] it is shown that T is measure isomorphic to K c .Hence if K c = F ◦ F ( N ) we have ϕ N is true if and only T ∼ = T − . Since S ∼ = T , ϕ N is true if and only S ∼ = S − . Thus if we define the realization function R by setting R ( K c ) = S , we see that R ◦ F ◦ F O is a reduction of the collection of codes for trueΠ -sentences to the set of recursive diffeomorphisms isomorphic to their inverses.This is the content of figure 1.In addition to these results in [13], we will show that the sequence of S n ’s canbe taken to be effective, converge in the C ∞ topology and that if S ( N ) comes from N and S ( M ) comes from M , then S ( N ) (cid:54)∼ = S ( M ). This will complete the proof ofTheorem 1. We encode the symbolic system K c on the torus by inductively constructing thesequence of h n ’s. The map h is the identity map corresponding to W c = Σ. Tobuild h n +1 , T is subdivided into rectangles which are then permuted. Definition 38 (Rectangular subdivisions) . Let n, m ∈ N . • For an arbitrary natural number q , I q represent the collection of intervals [0 , q ) , [ q , q ) , . . . , [ q − q , . • Given I q and I s , let I q ⊗ I s be the collection of all rectangles R = I × I ,where I ∈ I q and I ∈ I s . • Let D ⊆ T . Then, for a collection of rectangles ξ , the restriction of ξ to D isgiven by ξ (cid:22) D = { R ∩ D : R ∈ ξ } . • Recall the parameter sequences (cid:104) q n : n ∈ N (cid:105) and (cid:104) s n : n ∈ N (cid:105) . Further recallthat s n = |W cn | and q n = | u | for u ∈ W cn . Define ξ n = I q n ⊗ I s n . • Lastly, for ≤ i < q n and ≤ j < s n , let R ni,j be the element of ξ n given by [ iq n , i +1 q n ) × [ js n , j +1 s n ) . Note that there is a straightforward description of the action of R α n on ξ n : R α n : R ni,j (cid:55)→ R ni + p n ,j where addition in the subscript is performed modulo q n .The map h n +1 will be defined as a permutation of I k n q n ⊗ I s n +1 and thus inducesa permutation of ξ n +1 . It is important to make h n +1 commute with R α n . To dothis h n +1 is first defined on ( I k n q n ⊗ I s n +1 ) (cid:22) ([0 , /q n ) × [0 , T . 51 onstructing the h n ’s: The paper [13] is concerned with realizing circular sys-tems, and so builds the h n ’s in terms of the prewords used to construct the sequence (cid:104)W cn : n ∈ N (cid:105) . In the case that (cid:104)W cn : n ∈ N (cid:105) is in the range of F , the prewords aredetermined by the underlying odometer based sequence (cid:104)W n : n ∈ N (cid:105) . We describe h n +1 directly in terms of the odometer sequence (cid:104)W n : n ∈ N (cid:105) = F O ( N ).Fix enumerations (cid:104) w ns : 0 ≤ s < s n (cid:105) of each W n . The words in W n +1 areconcatenations of words in W n : w n +1 s = w w . . . w k n − where each w i = w ns (cid:48) for some s (cid:48) .To each w n +1 s associate the horizontal strip [0 , × [ s/s n +1 , ( s + 1) /s n +1 ) andeach w ns (cid:48) with [0 , × [ s (cid:48) /s n , ( s (cid:48) + 1) /s n ). Proposition 39.
There is a permutation of I k n q n ⊗ I s n +1 (cid:22) [0 , /q n ) × [0 , suchthat for all ≤ s < s n +1 ,if w i = w ns (cid:48) then h n +1 ([ i/k n q n , ( i + 1) /k n q n ) × [ s/s n +1 , ( s + 1) /s n +1 )) (22) ⊆ [0 , /q n ) × [ s (cid:48) /s n , ( s (cid:48) + 1) /s n ) . (cid:96) Equation 22 gives regions that each atom of I k n q n ⊗ I s n +1 (cid:22) [0 , /q n ) × [0 ,
1) mustbe sent to by h n +1 . To prove there is such a permutation we see that each regionhas exactly the same number of subrectangles as the cardinality of the collection ofatoms that must map into it.We count occurrences of n -words in ( n + 1)-words. Fix a word w n +1 s = w w . . . w k n − ∈ W n +1 . Then, by strong uniformity each n -word w ns (cid:48) occurs k n /s n times as a w i . So eachword w n +1 s puts k n /s n rectangles in a target region. Since there are s n +1 manywords of the form w n +1 s the target regions must contain s n +1 ( k n /s n ) rectangles.Each horizontal strip of [0 , ⊗I s n is divided into s n +1 /s n many horizontal stripsby [0 , ⊗ I s n +1 and each vertical strip of I q n ⊗ [0 ,
1) is divided into k n many verticalstrips by I k n q n ⊗ [0 , ξ n (cid:22) [0 , /q n ) × I s is dividedinto k n ( s n +1 /s n ) rectangles by I k n q n ⊗ I s n +1 . In particular I k n q n ⊗ I s n +1 (cid:22) [0 , /q n ) × [ s (cid:48) /s n , ( s (cid:48) + 1) /s n ) has k n ( s n +1 /s n ) many atoms.Hence each target region contains the same number of rectangles as atoms sentto it and there is a map h n +1 satisfying equation 22. (cid:97) Since h n +1 is a permutation of I k n q n ⊗I s n +1 (cid:22) [0 , /q n ) × [0 ,
1) for each 1 ≤ i < q n ,it can be copied onto each I k n q n ⊗ I s n +1 (cid:22) [ ip/q n , ( ip + 1) /q n ). The result of this isa permutation of I k n q n ⊗ I s n +1 (and hence ξ n +1 ) that commutes with the rotation R α n . Remark 40.
It is a clear that h n +1 can be defined in a primitive recursive wayusing the data W n +1 . emark It is shown in [13] that having defined the sequence of h n ’s in this manner,for sufficiently fast growing l n the transformations T n converge in measure to ameasure preserving transformation T : ( T , λ ) → ( T , λ ) that is isomorphic to theoriginal circular system defined by (cid:104)W cn : n ∈ N (cid:105) . The map taking (cid:104)W cm : m ≤ n (cid:105) to T n is primitive recursive. T n We now must smooth the T n ’s to produce S n ’s that have measure-theoretic limit S which is isomorphic to T . Secondly, we show that S is a recursive diffeomorphism.For our discussion of smoothing we need an effective complete metric on the C ∞ -diffeomorphisms. Note that the C ∞ topology is the coarsest common refinement ofthe C k topologies for each k ∈ N . There are many choices for effective/recursivemetrics generating the C k topology for each k ; for instance the metric derived fromthe norm given by (cid:107) f (cid:107) k = max x ∈ T (cid:107) f ( x ) (cid:107) + (cid:107) Df ( x ) (cid:107) + · · · + (cid:107) D k f ( x ) (cid:107) k where (cid:107)−(cid:107) j is the j -norm on R j +2 . Given an effective sequence of complete metrics (cid:104) d k : k ∈ N (cid:105) generating the C k topologies, with distances bounded by 1, then d ∞ = ∞ (cid:88) k =0 − ( k +1) d k generates the C ∞ topology.Fix such a complete effective metric giving rise to the C ∞ topology on T .Without loss of generality we can assume that d ∞ ( S, T ) ≤ max x ∈ T d T ( S ( x ) , T ( x )) , (23)where d T is the ordinary metric on T .To pass from the discontinuous Z n ’s to diffeomorphisms, the h i ’s are replaced bysmooth h si which are very close approximations and give the H n ’s in equation 20.Then the H n ’s will also be diffeomorphisms. While there is no control over the C ∞ -norms of the H n , the key observation at the heart of the Anosov-Katok methodis the following: if h sn +1 commutes with R α n then S n = H n ◦ R α n ◦ H − n = H n ◦ h sn +1 ◦ ( h sn +1 ) − ◦ R α n ◦ H − n = H n ◦ h sn +1 ◦ R α n ◦ ( h sn +1 ) − ◦ H − n = H n +1 ◦ R α n ◦ H − n +1 . (24)53ence by taking α n +1 sufficiently close to α n , S n +1 can be taken as close as necessaryto S n in the C ∞ -norm.To carry out this plan we begin by describing how we smooth the h n ’s. This isdone explicitly in Theorem 35 of [13], which says: Theorem 41 (Smooth permutations) . Let T be divided into the collection of rect-angles I n ⊗ I m and choose (cid:15) > . Let σ be a permutation of the rectangles. Thenthere is an area preserving C ∞ -diffeomorphism ϕ : T → T such that ϕ is theidentity on a neighborhood of the boundary of [0 , × [0 , and for all but a set ofmeasure at most (cid:15) , if x ∈ R , then ϕ ( x ) ∈ σ ( R ) for all R ∈ I n ⊗ I m . In Lemma 36 of [13] it is shown that an arbitrary permutation of I n ⊗ I m can bebuilt by taking a composition of transpositions of adjacent rectangles. The trans-formation ϕ is then built effectively as a composition of smooth near-transpositionsthat swap adjacent rectangles. We summarize the proof. More details appear inAppendix D, where it is shown that it can be carried out recursively in a code forthe permutation σ .The main technical point for building the near-transpositions of adjacent rect-angles is captured by showing that for all 0 < γ < (cid:15) < − γ ,there is a diffeomorphism ϕ of the unit disk in R such that:1. ϕ rotates the top half of the disk of radius γ to the bottom half and viceversa.2. ϕ is the identity in a neighborhood of the unit circle of width less than (cid:15) .The map ϕ is constructed by considering a primitive recursive C ∞ map f : [0 , → [0 , π ] that is identically equal to π on [0 , γ ] and is 0 in a neighborhood of 1. Then ϕ rotates the circle of radius r by f ( r ) radians. Taking γ very close to 1 gives asmooth near transposition.Using Riemann mapping theorem techniques, these rotations of the disk can becopied over to measure preserving maps from [ − , × [0 ,
1] to itself that1. take all but 1 − (cid:15)/ − , × [0 ,
1] to [0 , × [0 ,
1] and vice versa,2. are analytic on the interior of [ − , × [0 , { , , . . . mn } can be written as a composition of lessthan or equal to ( nm ) transpositions of the form ( k, k + 1), given any σ we canbuild ϕ by taking (cid:15) small enough and composing sufficiently good approximationsbetween adjacent rectangles corresponding to the transpositions composed to create σ . 54 uilding S n . Using Theorem 41 we can effectively choose a smooth h sn +1 whichwell-approximates h n +1 measure theoretically. By choosing the approximation well,we can guarantee that the S n in equation 21 moves the partitions ξ n very close towhere the T n ’s move the ξ n ’s.Since h sn +1 is effective, using the continuity of composition with respect to d ∞ , S n +1 can be made arbitrarily close to S n by taking α n +1 sufficiently close to α n .Thus if α n converges to α sufficiently quickly, the sequence (cid:104) S n : n ∈ N (cid:105) is Cauchywith respect to the complete metric d ∞ and hence converges to a smooth measurepreserving diffeomorphism S . Taking the sequence of h sn ’s to be sufficiently closeto the h n ’s the S n ’s are sufficiently close to the T n ’s to apply Lemma 30 of [13] toshow that the diffeomorphism S is measure theoretically isomorphic to T . Hence( T , λ, S ) is measure theoretically isomorphic to ( K c , ν, Sh ). The induction.
The discussion above was predicated on choosing the l n ’s togrow fast enough . We now show how to inductively choose lower bounds l ∗ n onthe l n . Numerical Requirement E gives one collection of lower bounds for the l ’s,independently of the choices of the maps H n and numbers α n . Hence we choose l ∗ n to dominate this sequence of lower bounds, as well as the lower bounds we add here.Suppose we have defined H n +1 from h sn +1 and H n in a manner that satisfiesequation 24 holds and that the H n ’s can be computed effectively. Then, for anygiven (cid:15) , and small rational β , d ∞ ( H n +1 R α n + β H − n +1 , H n R α n H − n ) (25)can be primitively recursively computed to within a given (cid:15) . Moreover, this is adecreasing function of β for small β >
0. Thus one one can effectively find a δ suchthat if | α n +1 − α n | < δ , then d ∞ ( S n +1 , S n ) < − ( n +1) .Recall the definitions of the α n = p n /q n from equations 13, 14 and 15. Then α n does not depend on l n and α n +1 = α n + 1 /k n l n q n . Thus to make α n +1 close to α n it suffices to make l n sufficiently large that1 /k n l n q n < δ. (26) Numerical Requirement G
The parameter l n is chosen sufficiently large that d ∞ ( S n +1 , S n ) < − ( n +1) (27)The numbers α n , (cid:104)W cm : m ≤ n (cid:105) and (cid:104) h sm : m ≤ n + 1 (cid:105) determine the δ inequation 26 and thus how large l n must be. All of this data can be computedrecursively from (cid:104)W m : m ≤ n + 1 (cid:105) . (We note that neither the choice of s n +1 northe definition h sn +1 uses l n .) 55 .3.3 The effective computation of S n We now show that each element of the sequence (cid:104) S n : n ∈ N (cid:105) is effectively com-putable (Definition 7). Claim 42.
The functions h sn and R α n are effectively computable C ∞ -functions. Asa consequence each S n is effectively uniformly continuous.Proof of Claim 42. For simplicity of exposition, we only show how to compute themodulus of continuity and approximation for S n itself; finding the modulus of con-tinuity and approximations to the higher differentials is conceptually identical butnotationally cumbersome.Recall that we must produce two functions: • A modulus of continuity, d : N → N , and • An approximation, f : ( { , } × { , } ) < N → ( { , } × { , } ) < N .It is routine to check that if T and T are effectively uniformly continuous—that is, if there exist moduli of continuity d and d and approximations f and f corresponding to each—then the composition, T ◦ T is effectively uniformlycontinuous.The second part of the claim follows from the first since Equations (21) and (20)show, S n = h s ◦ h s ◦ · · · ◦ h sn ◦ R α n ◦ ( h sn ) − ◦ · · · ◦ ( h s ) − ◦ ( h s ) − . (28)The case of R α n is particularly simple. Since R α n is an isometry, it has aLipschitz constant of 1. In particular, the modulus of continuity is simply given by d ( n ) = n , and, since R α n is well-defined on rational points, we can also determinethe approximation by setting f to be([ x ] m , [ y ] m ) (cid:55)→ ([ x ] m + [ α n ] m , [ y ] m )Where [ z ] m denotes the smallest dyadic rational k × − m for 0 ≤ k ≤ m minimizing | z − [ z ] m | .In the case of h sm for m ≤ n , recall from the discussion after Theorem 41 that h sm can be built as a composition of a sequence of smooth transpositions: h sm = σ s ◦ σ s ◦ · · · ◦ σ st ( m ) Note that the number of transpositions necessary, t ( m ) < | ξ m +1 | = ( k m · q m · s m +1 ) ,and is a computable function of m since it is the number of transpositions necessaryto build the permutation in Proposition 39.Since σ sj is a smooth transposition of an explicit form (given in Appendix D),one can calculate a uniform Lipschitz constant L sj for it; hence, taking L m > max s ≤ t ( m ) L sj , we have that | h sm ( x ) − h sm ( y ) | < ( L m ) t ( m )+1 | x − y | . h sm is given by d ( n ) = n + (cid:100) t ( m ) · log ( L m ) (cid:101) , (29)where (cid:100) x (cid:101) is the smallest integer greater than x . The construction of a primitiverecursive approximation to h sm is straightforward from the primitive recursive ap-proximations to the σ sn ’s. As we remarked in Section 1.4, it follows that ( h sm ) − isprimitive recursive. (cid:97) In summary, the modulus of continuity and approximation for S n can be calcu-lated using the following steps:1. Compute the (cid:104) h m : m ≤ n (cid:105) ;2. Build the approximations to h sm and ( h sm ) − using h m and the smooth trans-positions given in Appendix D for m ≤ n ;3. Compute the moduli of continuity of (cid:104) h sm : m ≤ n (cid:105) and their inverses;4. Compute (cid:104) α m : m ≤ n (cid:105) (and, consequently, the approximations and moduli ofcontinuity for (cid:104)R α m : m ≤ n (cid:105) );5. Compute the approximation and modulus of continuity of S n by composingthe approximations and moduli of continuity calculated in Steps 1 through 4according to Equation (28). Theorem 1 claims the existence a computable function F , which on inputting anatural number N (corresponding to the Π sentence ϕ ) outputs a code for a com-putable diffeomorphism S ( N ) of T . Whether or not S ( N ) is measure theoreticallyconjugate to S ( N ) − is equivalent to the truth of falsity of ϕ . Finally for differentnumerical inputs the corresponding S ’s will not be isomorphic. In summary, letting S = S ( N ),(A) If N codes ϕ , then ϕ is true if and only iff S ∼ = S − (B) On input N , F recursively determines a code for an effectively C ∞ map of thetorus to itself, i.e., F determines:i.) A computable function d : N × N → N , where d ( k, − ) computes the moduliof uniformity of the k th differential of S ( N ), andii.) A computable function f ( k, − ) where f ( k, − ) is a map on dyadic rationalpoints of T approximating D k S ( N ) (the k -th differential of S ( N )). Givenan input that is precise to d ( k, n ) digits, f ( k, − ) approximates the first n -partial derivatives { ∂ n ∂ i x∂ n − i y : 0 ≤ i ≤ n } to n -digits.57C) If N (cid:54) = M , then the associated diffeomorphisms S ( N ) and S ( M ) are not con-jugate.Because the function F maps natural numbers to natural numbers (the codes for thediffeomorphisms) we let F (cid:91) be the associated function R ◦ F ◦ F O that maps into thespace of actual diffeomorphisms. It produces a diffeomorphism of T from a G¨odelnumber N for a Π set. We show F (cid:91) satisfies ( A ) and ( C ) and then argue there isa (primitively) computable routine coded by F ( N ) that has the same values. Item (A)
Given N , F O ( N ) computes an odometer-based construction sequence (cid:104)W n : n ∈ N (cid:105) . By Theorem 9, if K ( N ) is the uniquely ergodic symbolic shiftassociated with the construction sequence then K ( N ) ∼ = K ( N ) − if and only if ϕ istrue.The sequences (cid:104) l n : n ∈ N (cid:105) , (cid:104)W cn : n ∈ N (cid:105) and (cid:104) h sn : n ∈ N (cid:105) are computed. If K c is the circular system associated with (cid:104)W cn : n ∈ N (cid:105) , then Proposition 35 shows that K c ∼ = ( K c ) − if and only if ϕ is true.Finally the realization map R preserves isomorphism. So if S = R ( K c ), then( T , λ, S ) ∼ = ( T , λ, S − ) if and only if ϕ is true. Item (C)
We need to see that for
M < N, S ( N ) (cid:54)∼ = S ( M ). Since the realizationmap R preserves isomorphism if suffices to see that ( K c ) M = F ◦ F O ( M ) is notisomorphic to ( K c ) N .By Corollary 34 we see that the Kronecker factor of K M is K α M . Any isomor-phism between ( K c ) N and ( K c ) M must take the respective Kronecker factor of oneto the other, hence would imply an isomorphism between K α N and K α M .However this is impossible since Corollary 34 implies that π > α M > α N > Item (B) F (cid:91) is a map from N to diffeomorphisms. By the result of Section 3.3.3,the diffeomorphisms are recursive. We must show that there is a recursive algorithmcoding a function F that computes the moduli of continuity and approximations toeach F (cid:91) ( N ) and its differentials.We use the notation d N to denote the modulus of continuity returned by F ( N ),and we use the notation f N to denote the approximation. Without loss of generality,we restrict our attention to d (0 , − ) and f (0 , − )—that is, the C modulus of conti-nuity and approximation of S . The calculation for d ( k, − ) and f ( k, − ) for k > d N ( n ) for d (0 , n ) and f N ( (cid:126)s, (cid:126)t ) for f (0 , (cid:126)s, (cid:126)t ).Let us first consider the modulus of continuity. The routine for computing d N depends on choosing a large number of numerical parameters: (cid:15) n , e ( n ) , s n , k n , l n , P N . a n (cid:29) b n or a n (cid:28) b n . It is routine that these can be satisfied in a primitive recursive manner–provided that the dependencies are consistent. This is verified in Appendix A whereit is shown that the dependencies among these constants form a directed acyclicgraph.The subroutine we describe next comes during the computation of F ( N ), andhence we may assume that we have the coefficients k n ( N − , l n ( N −
1) alreadycomputed. This computation was made during the first n steps of the computationof F ( N − ≤ m ≤ n + 1, make the following calculations, which recursivelydepend on smaller m . Specifically, W m is built from W m − using the SubstitutionLemma (Proposition 20) as described in section 2.6. Then h sm is built from theinformation in the words in W m . This allows l m to be chosen large enough thatNumerical Requirement G holds. This in turn defines p m and q m and allows W cm tobe built.The algorithm is illustrated in Figure 4.1. Using (cid:104) (cid:15) k : k ≤ m (cid:105) and W m − , choose k m large enough to satisfy the NumericalRequirements C and D and apply the Substitution Lemma m + 1 times togenerate W m ;2. Build h m , smooth it to get h sm and hence H m . Calculate H m ’s modulus ofcontinuity.3. Choose l m sufficiently large that Numerical Requirements E and F hold (with n + 1 = m ).4. Build W cm .5. Calculate the approximation and modulus of continuity corresponding to S m using the methods of Section 3.3.3.6. Continue until m = n + 1 and d S N ( n + 1), the n + 1 st approximation to themodulus of continuity of S n +1 is determined.7. Output d ( n + 1)( n + 1) = d S N ( n + 1).At the end, using the modulus of continuity d corresponding to S n +1 , output d N ( n ) = d (0 , n + 1) where d ( n + 1) is the modulus of continuity of S n +1 .To verify that this procedure actually yields a modulus of continuity for S , recallthat by Numerical Requirement G, Equation (27), it follows that d ∞ ( S n +1 , S ) < − ( n +1) .
59y inequality 23,max x ∈ T d T ( S n +1 ( x ) , S ( x )) ≤ d ∞ ( S n +1 , S ) < − ( n +1) . Since d ( n + 1) yields the number of digits of input necessary to approximate S n +1 to an accuracy of 2 − ( n +1) , it follows that the approximation of S n +1 is itself anapproximation of S which, given d ( n + 1) digits of binary input, is accurate towithin 2 − n .The approximation f for S is calculated almost identically, except for the output.Given ([ x ] d N ( n ) , [ y ] d N ( n ) ) ∈ ( { , } × { , } ) d ( n ) the output is f N ( n + 1) = (cid:16) f ([ x ] d N ( n ) ) , f ([ y ] d N ( n ) ) (cid:17) where f ( n + 1) = ( f , f ) is the approximation of S n +1 produced in Step 5, again inthe notation that [ z ] m is a m -digit binary approximation of z . Appendices
Appendices B-D were originally written by Johann Gaebler. The author has cleanedup exposition and done his best to fix various mistakes that appeared in Gaebler’soriginal version.
A Numerical Parameters
A.1 The Numerical Requirements Collected.
In this appendix we review the requirements on the numerical parameters used inthe construction. Specifically, in constructing the diffeomorphism F ( N ) we buildconstruction sequences (cid:104)W n : n ∈ N (cid:105) , (cid:104)W cn : n ∈ N (cid:105) that depend on N and realize thecorresponding circular system K c as a diffeomorphism. These steps are intertwined–for example the circular system is built as a function of the sequence (cid:104) k n , l n : n ∈ N (cid:105) .In turn the l n are chosen as function of (cid:104)W m : m ≤ n (cid:105) in order to facilitate thesmooth construction. To rigorously complete the proof we need to review all ofthese parameters and see that the inductive choices can be made consistently in aprimitive recursive way.At many stages in this paper we appeal to results from [11]. Hidden in thoseappeals is a sequence of parameters (cid:104) µ n : n ∈ N (cid:105) that is not explicitly mentioned inthe construction presented here. For this reason this review includes the inductiveconstruction of (cid:104) µ n : n ∈ N (cid:105) .A substantial difference between this paper and earlier constructions is that thedomain of the reductions in [9] and [11] is the space of trees of finite sequences ofnatural numbers. The analogue in this paper is that the only trees considered here60nput n and set m = 0. Initialize s m , (cid:104) ε m (cid:105) and (cid:104) (cid:15) m (cid:105) .Output d ( n + 1)( n + 1),where d ( n + 1) is S n +1 ’smodulus of continuity. Is m > n + 1? Choose Q m , ε m , µ m , (cid:15) m , s m +1 satisfying the Numer-ical Requirements.Calculate the modulusof continuity andapproximation of S m . Set m to m + 1. Calculate k m ≥ k m ( N −
1) and large enough forSubstitution LemmaChoose l m ≥ l m ( N − d ∞ ( S m , S m − ) < − m − , Build W cm . Build h m , h sm , H m andcalculate modulusof continuity of H m . Build W m noyesFigure 4: The algorithm for calculating the modulus of continuity d N of S . Thisalgorithm is easily altered to produce the approximation f N of S simply by changingthe output to ( f ( (cid:126)x ) , f ( (cid:126)y )), where f ( n + 1) = ( f , f ) is S n +1 ’s approximation.61re the trees of sequences (cid:104) (0 , , . . . n ) : n < Ω (cid:105) for Ω finite or infinite depending onthe input N . We note that these trees are really “stalks” and are finite or infinitedepending on Ω. Since the trees we consider in this paper are of this very specialform, the requirements are easier to satisfy.Closely following section 10 of [11] we begin with a review of the inductive re-quirements from [9]. We give them in the notation of [11]. These inductive require-ments are modified and simplified in the construction in the current manuscript.We note the versions used in this paper. Requirements that were instituted in [9] and their modifications.
Theserequirements were dubbed
Inherited Requirements in [11]. Requirements that werenew in [11] are called simply
Numerical Requirements , and requirements that ex-plicitly mentioned in the text of the paper are labelled with capital letters A-F.Recall that the number of elements of W m is denoted s m ; the numbers Q ms and C ms denote the number of classes and sizes of each class of Q ms respectively. Fromthe construction in [9] we have sequences (cid:104) (cid:15) n : n ∈ N (cid:105) , (cid:104) s n , k n , e ( n ) : n ∈ N (cid:105) . Inherited Requirement 1 (cid:104) (cid:15) n : n ∈ N (cid:105) is summable. Inherited Requirement 2 e ( n − the number of Q ni +1 classes inside each Q ni class. The numbers e ( n ) will be chosen to grow fast enough that2 n − e ( n ) < (cid:15) n (30)Similarly we set C nn = 2 e ( n ) as well. Modification:
In this paper the construction is simplified so to build W n +1 we have exactly n + 2-substitutions of each of size 2 e ( n ) . Hence we can replacethis requirement by the simple formula s n = 2 ( n +1) e ( n − . In particular s m , Q mi and C mi are all to be powers of 2. Inherited Requirement 3
For all n ,2 (cid:15) n s n < (cid:15) n − (31) Inherited Requirement 4 (cid:15) n k n s − n − → ∞ as n → ∞ (32) Inherited Requirement 5 (cid:89) n ∈ N (1 − (cid:15) n ) > Comment:
Since this is equivalent to the summability of the (cid:15) n -sequence, itis redundant and we will ignore in the rest of this paper62 nherited Requirement 6 (Original Version) There will be prime numbers p i such that K i = p i s i − K i − (i.e. k i = p i s i − ). The p n ’s grow fast enoughto allow the probabilistic arguments in [9] involving k n to go through. Modification
For all n , k n = P N (cid:96) s n , where for each n , (cid:96) is large enough forthe substitution argument involving k n to go through. Comment:
In [9] K n was a product of a sequence of prime numbers. Therequirement on the sequences of prime numbers was that they were almostdisjoint for different trees and that they grew sufficiently quickly. In thispaper K N is P N ∗ (cid:96) for a large (cid:96) .Since we have only one collection of very special trees the requirement sim-plifies to needing that the (cid:96) in the exponent grows sufficiently quickly for theSubstitution Lemma (Proposition 20) argument to work. Inherited Requirement 7 s n is a power of 2. Comment:
This again is redundant as the modified Inherited Requirement 2says directly that s n = 2 ( n +1) e ( n − Inherited Requirement 8
For all n , (cid:15) n < − n . Numerical Requirements introduced in [11]Numerical Requirement 1 l n > ∗ n and 1 /l n − > (cid:80) k = n /l k . Numerical Requirement 2 (cid:104) ε n : n ∈ N (cid:105) is a sequence of numbers in [0 ,
1) suchthat ε N > (cid:80) n>N ε n . Numerical Requirement 3
For all n , ε n − > sup n Set t n = min( ε n , /Q n ) and take0 < µ n < t n min k ≤ n − n − (cid:18) t k (cid:19) . Then for all m t m > ∞ (cid:88) n = m µ n t n . This is sufficient to carry out the various arguments, for example using theBorel-Cantelli Lemma. 63 umerical Requirement 5 (cid:80) | G n | Q n < ∞ . Modification In this paper case | G n | ≤ (cid:88) n Q n < ∞ . Comment: Since Q n = e ( n ) and Requirement 2 implies that 2 n − − e ( n ) → Numerical Requirement 6 l n is big enough relative to a lower bound determinedby (cid:104) k m , s m : m ≤ n (cid:105) , (cid:104) l m : m < n (cid:105) and s n +1 to make the periodic approxima-tions to the diffeomorphism F ( N ) converge. Numerical Requirement 7 s n goes to infinity as n goes to infinity and s n +1 is apower of s n . Comment Since s n is a power of 2 e ( n ) and e ( n ) → ∞ , this is trivial. Numerical Requirement 8 s n +1 ≤ s k n n . Numerical Requirement 9 The (cid:15) n ’s are decreasing, (cid:15) < / 40 and (cid:15) n < ε n . Numerical Requirement 10 k n is chosen large enough relative to the lower bounddetermined by s n +1 , (cid:15) n to apply the Substitution Lemma and construct thewords in W n +1 . Implicitly this requires that 1 /k n < (cid:15) n / Comment: This is essentially the same as Inherited Requirement 6. Numerical Requirement 11 (cid:15) n is small relative to µ n . Modification Remark 94 of [11] says that for quantities r ( x, y ) , r ( x, C ) , f ( x )that are determined by counting occurrences of x, y in words in an alphabet L with s letters that have a given length (cid:96) . It says for all (cid:15) > δ such that if for all x, y ∈ L , (cid:12)(cid:12)(cid:12)(cid:12) r ( x, y ) (cid:96) − s (cid:12)(cid:12)(cid:12)(cid:12) < δ then for all x : (cid:12)(cid:12)(cid:12)(cid:12) r ( x, C ) f ( x ) − Cs (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) From the proof of the lemma it is straightforward to find an explicit formulafor δ ( (cid:15), | C | , (cid:96), s ) for an upper bound on δ . The small relative clause can berephrased as asking that (cid:15) n < δ ( µ n , C n , q n , s n ) . Since C n = 2 − e ( n ) s n , δ is really a function of µ, q n , s n .64 umerical Requirement 12 (cid:15) k > 20, the (cid:15) n k n ’s are increasing and (cid:80) /(cid:15) n k n < ∞ . Numerical Requirement 13 The numbers (cid:15) n should be small enough, as a func-tion of Q n , that for all w , w ∈ W cn +1 ∪ Rev ( W cn +1 ) with [ w ] (cid:54) = [ w ] thefollowing inequality holds:¯ d ( w (cid:22) I , w (cid:22) I ) > (1 − /Q n ) γ n . where the γ i ’s are defined inductively as: γ = (1 − / − (cid:15) )(1 − /(cid:15) k )(1 − /l )for n ≥ γ n = γ (cid:89) In this paper we have some supplemental numerical requirements. We list onlythose that are not redundant given the requirements listed above. Numerical Requirement B k n ( N − ≤ k n ( N ) . Numerical Requirement D /k n < (cid:15) n / . Numerical Requirement E l n ( N − ≤ l n ( N ) . Numerical Requirement F (cid:80) n k n < ∞ . Numerical Requirement G d ∞ ( S n +1 , S n ) < − ( n +1) . A.2 Resolution A list of parameters, their first appearances and their constraints We classify the constraints on a given sequence according to whether they referto other sequences or not.Requirements that inductively refer to the same sequence are straightforwardlyconsistent and can be satisfied with a primitive recursive construction. For examplea requirement that a certain inductively sequence involving a given variable besummable is satisfied by asking that the n th sequence be less than 2 − n . We callthese absolute conditions.Those requirements that refer to other sequences risk the possibility of beingcircular and thus inconsistent or not being computable from the data in the othersequences. We refer to these conditions as dependent conditions.65. The sequence (cid:104) k n : n ∈ N (cid:105) . Absolute conditions: A1 The sum (cid:80) n n /k n is finite. A2 k = P N and k n ( N ) ≥ k n ( N − D1 Numerical Requirement 10, is a lower bound for k n depends on s n +1 , (cid:15) n ,asking that k n be large enough for the word construction using the Sub-stitution Lemma to work. Why is this primitive recursive? Given s n +1 and (cid:15) n , the discussion in theproof of Lemma 22 shows that a lower bound for k n can be given fromHoeffing’s Inequality (Theorem 21) in a primitive recursing way. So theonly possible issue is circularity. D2 Inherited Requirement 6. In this context it says that k n = P N (cid:96) s n for alarge (cid:96) . Why is this primitive recursive? k n is defined in equation 11 where itexplicitly is a multiple of s n . The size of (cid:96) is determined by D1 . D3 From Inherited Requirement 4, equation 32 requires that (cid:15) n k n s − n − goesto ∞ as n goes to ∞ . Why is this primitive recursive?: This can be satisfied primitively recur-sively by choosing k n to be an integer larger than s n (cid:15) n .We note that equation 32 implies that (cid:80) /(cid:15) n k n is finite. D4 Numerical Requirement 8 implies that k n is large enough that s n +1 ≤ s k n n . Comment: This is easily satisfied by taking k n ≥ log ( s n +1 ) log ( s n ) . D5 Numerical Requirement D says 1 /k n < (cid:15) n / Comment: As long as (cid:15) n is defined before k n , Requirement D is immediate by taking k n > /(cid:15) n . D6 Numerical Requirement 12 says that (cid:15) k > 20 and the (cid:15) n k n ’s are increas-ing and (cid:80) /(cid:15) n k n is finite. Why is this primitive recursive? As noted the last condition follows fromD3. The other parts of Numerical Requirement 12 are satisfied primitiverecursively by taking k n to be an integer at least n(cid:15) n .From D1-D5, we see that k n is dependent on the choices of (cid:104) k m , l m : m The (cid:107) (cid:107) ∞ -norm of S ◦ T can be computedeffectively from the (cid:107) (cid:107) ∞ -norms of S and T . In particular there is a primitivelyrecursively computable real number M such that d ∞ ( S n +1 , S n ) < M | α n +1 − α n |≤ Mq n +1 . and the latter inequality is from equation 15. Since q n +1 = k n l n q n we get anexplicit lower bound on l n .3. The sequences (cid:104) s n : n ∈ N (cid:105) and (cid:104) e ( n ) : n ∈ N (cid:105) . We treat these sequencesas equivalent since s n = 2 ( n +1) e ( n − .Absolute conditions A5 Inherited Requirement 7 says that s n is a power of 2. A6 The sequence s n goes to infinity. A7 s n +1 is a multiple of s n .Dependent conditions D8 The function e ( n ) : N → N referred to in equation 30 gives the numberof Q ns +1 classes inside each Q ns class. It has the dependent requirementthat 2 n − e ( n ) < (cid:15) n − . Moreover s n = 2 ( n +1) e ( n ) .The result is that s n +1 depends on the first n + 1 elements of T (cid:104) k m , s m , l m : m < n (cid:105) , s n , and (cid:15) n . Why is primitive recursive? The only requirement for choosing s n +1 is that2 − e ( n +1) < (cid:15) n − n and this is clearly primitively recursively satisfiable. It is important to observe that the choice of s n +1 does not depend on k n or l n . The sequence (cid:104) (cid:15) n : n ∈ N (cid:105) . Absolute conditions A8 Numerical Requirement 9 and Inherited Requirement 1 say that the (cid:104) (cid:15) n : n ∈ N (cid:105) is decreasing and summable and (cid:15) < / A9 Inherited Requirement 8 says that (cid:15) n < − n Dependent conditions D9 Numerical Requirement 9 says (cid:15) n < ε n . D10 Equation 31 of Inherited Requirement 3 says 2 (cid:15) n s n < (cid:15) n − D11 Numerical Requirement 11 says that (cid:15) n must be small enough relativeto µ n . D12 Numerical Requirement 13 says that (cid:15) n is small as a function of Q n .The result is that (cid:15) n depends exogenously on the first n elements of T , andon Q n , s n , ε n , (cid:15) n − and µ n . Why is this primitive recursive? The only issue might be D10, but this is ex-plicitly solved in Numerical Requirement 11, which describes how to calculatean explicit function δ ( µ n , q n , s n ) such that Numerical Requirement 11 holds if (cid:15) n < δ ( µ n , q n , s n ).5. The sequence (cid:104) ε n : n ∈ N (cid:105) . Absolute conditions A10 Because the sequence (cid:104) q n : n ∈ N (cid:105) is increasing Numerical Requirement 3is satisfied if ε n − > (cid:80) k ≥ n ε n , a rephrasing of Numerical Requirement 2.This is an absolute condition and implies that (cid:104) ε n : n ∈ N (cid:105) is decreasingand summable.Dependent conditions D13 (cid:104) k n ε n : n ∈ N (cid:105) goes to infinity. This already follows from the fact that (cid:15) n < ε n and item D4 .Since item D12 follows from item D4, all of the requirements on (cid:104) ε n : n ∈ N (cid:105) are absolute or follow from previously resolved dependencies. Moreover theyare trivial to satisfy primitively recursively.6. The sequence (cid:104) Q n : n ∈ N (cid:105) . Recall Q n is the number of equivalence classes in Q n . We require:Absolute conditions 68 The only requirement on the choice of Q n not accounted for by thechoices of the other coefficients is that (cid:80) /Q n < ∞ . Dependent conditionsNone7. The sequence (cid:104) µ n : n ∈ N (cid:105) . This sequence gives the required pseudo-randomness in the timing assump-tions.Absolute conditionsNone.Dependent conditions D14 Numerical Requirement 4 appearing in this paper is written explicitlyas follows: Set t n = min( ε n , /Q n ) and take0 < µ n < t n min k ≤ n − n − (cid:18) t k (cid:19) . Then for all m t m > ∞ (cid:88) n = m µ n t n . The recursive dependencies of the various coefficients are summarized in Figure 5,in which an arrow from a coefficient to another coefficient shows that the latteris dependent on the former. Here is the order the the coefficients can be chosenconsistently. Assume: The coefficient sequences (cid:104) k m , l m , Q m , µ m , (cid:15) m , ε m : m < n (cid:105) and s n havebeen chosen. To do: Choose k n , l n , Q n , µ n , (cid:15) n , ε n and s n +1 . Each requirement is to choosethe corresponding variable large enough or small enough where theseadjectives are determined by the dependencies outlined above.Figure 5 gives an order to consistently choose the next elements on the sequences;Choose the successor coefficients in the following order: Q n , ε n , µ n , (cid:15) n , s n +1 , k n , l n . n ε n µ n (cid:15) n s n +1 k n l n (cid:63)(cid:64)(cid:64)(cid:64)(cid:82) (cid:63)(cid:0)(cid:0)(cid:0)(cid:9)(cid:63)(cid:63)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:65)(cid:85)(cid:64)(cid:64)(cid:64)(cid:82) Figure 5: Order of choice of Numerical parameters dependency diagram. B Logical Background B.1 Logical Basics What follows is a very brief and relatively informal introduction to basic first-orderlogic and the formal language L PA , the language of first-order (or Peano ) arithmetic .For a more thorough treatment, see [8]. B.1.1 The language L PA The language of first-order arithmetic , L PA , is made up of the following pieces:1. It has variables x , x , . . . x n , . . . , two constant symbols 0 , 1, a relation symbol < and two function symbols + , ∗ .2. Terms: Terms are expressions in L PA that are intended to correspond toactual objects (i.e., numbers). The most basic terms are variables (which wefrequently denote with the informal symbols “ x ”, “ y ”, “ z ”, etc.) and constants ,namely “0” and “1”. New terms can be built inductively using old terms byapplying function symbols to them. Formally: if τ . . . τ n are terms and f is an n -place function symbol then f τ . . . τ n is a term. Since our function symbolsare binary the new term would either be + τ τ or ∗ τ τ .70t is necessary to prove unique readability for terms–that from the sequence ofsymbols constituting the term one can uniquely recapture its inductive con-struction. Frequently one drops the formal definition and inserts parenthesisto make the terms humanly readable. For example instead of writing + x x + 1). Or instead of writing ∗ + x x x + 1) ∗ ( x + 2)). However the first is the formally correct stringof symbols constituting the term.Thus, “ x ” , “ y ”are terms, as is “(( x + y ) + 1) ∗ y ”For n a natural number, we often write n as though it were a term, but thisis an abbreviation for the term n -times (cid:122) (cid:125)(cid:124) (cid:123) · · · + 1(with parenthesis suitably added). So n is a term in the language L PA , not anatural number in N .)It is important to note that all polynomials with natural number coefficientscan be expressed as terms.3. Atomic formulas: An atomic formula is an expression of the form“ t = t ”or “ t < t (cid:48)(cid:48) where t and t are terms.4. Compound formulas: A compound formula —also called a well-formed for-mula or, more simply, a formula —is defined recursively as follows:(a) An atomic formula is a formula.(b) If F and F are formulas, then “ ¬ F ”, “ F ∧ F ”, “ F ∨ F ”, “ F → F ”,etc., are formulas. (Here, “ ∧ ” represents and , “ ∨ ” represents (inclusive)or , “ → ” represents implies , and so on.)(c) If F is a formula, then “( ∀ u ) F ” and “( ∃ u ) F ” are formulas, where u hereis any variable. (These quantifiers have their usual meanings: “ ∀ ” canbe read “for all” and “ ∃ ” can be read “there exists.”)71 variable is free if it doesn’t appear inside the scope of a quantifier; otherwise, avariable is bound . To illustrate, the variable x is free in the formula “ x = x ” butbound in the formula “( ∀ x )( x = x )”.A sentence is a formula with no free variables. For instance, “ x + y = y + x ” is a well-formed formula, but not a sentence. However, its universal closure“( ∀ x )( ∀ y )( x + y = y + x )” is a sentence.Whether the variable x is “really” universal—i.e., within the scope of the quan-tifier ( ∀ x )—or existential—i.e., within the scope of the quantifier ( ∃ x )—can be aslightly subtle matter. For instance, the following two formulas are equivalent:“[( ∀ x )( y ∗ x = y )] → (0 = 1)”and “[( ∃ x ) ¬ ( y ∗ x = y )] ∨ (0 = 1)”Note that these are both equivalent to the formula“( ∃ x )[ ¬ ( y ∗ x = y ) ∨ (0 = 1)]”In the last formula the quantifier is ahead of all Boolean Combinations. B.1.2 Bounded quantifiers It is convenient to think of L PA as having two different kinds of quantifiers: un-bounded quantifiers —e.g., ( ∀ x, y, z )—and bounded quantifiers —e.g., ( ∃ x < in-crease the logical complexity of a formula, whereas bounded quantifiers leave thelogical complexity of a formula the same.The clearest manifestation of this is the fact that the order of unbounded quan-tifiers cannot be changed without changing the meaning of a formula, while, ingeneral, bounded quantifiers can be pushed past unbounded quantifiers, possibly atthe cost of introducing additional unbounded quantifiers, but without altering thetruth or falsity of the formula in question. This is a consequence of working in thesetting of Peano Arithmetic.For instance, “( ∀ x )( ∃ y )( x = 0 ∨ x = y + 1)”is a true statement about the natural numbers, whereas“( ∃ y )( ∀ x )( x = 0 ∨ x = y + 1)”is a false statement. However,“( ∀ x < ∃ y )( x = 0 ∨ x = y + 1)”72s equivalent to “( ∃ z )( ∀ x < ∃ y < z )( x = 0 ∨ x = y + 1)” , which has the unbounded quantifier in the outermost position. In this examplethe two sentences are trivially equivalent because they are both true. However thisquantifier interchange would preserve true no matter what is inside the scope of thequantifiers.Formulas containing only bounded quantifiers are known as ∆ -formulas. A∆ -formula can be rewritten as a Boolean combination of polynomials. For instance“( ∀ x < ∃ y < x + y = z )”can be rewritten in the less compact but equivalent form“ [(0 + 0 = z ) ∨ (0 + 1 = z ) ∨ (0 + z = z )] ∧ [(1 + 0 = z ) ∨ (1 + 1 = z ) ∨ (1 + z = z )] ∧ [( z + 0 = z ) ∨ ( z + 1 = z ) ∨ ( z + z = z )] (cid:48)(cid:48) . B.1.3 Formula complexity As was the case for the expression“[( ∀ x )( y ∗ x = y )] → (0 = 1)”every formula in L PA can be rewritten in an equivalent form where all of the quan-tifiers are at the beginning. As was noted in the previous section, we can also ensurethat any bounded quantifiers appear after all of the unbounded quantifiers. Thus,we can ensure that every formula has the following bipartite structure: (1) alternat-ing blocks of unbounded “ ∀ ” and “ ∃ ” quantifiers (not necessarily in that order); (2)a ∆ -formula afterward. For instance,“ unbounded quantifiers (cid:122) (cid:125)(cid:124) (cid:123) ( ∀ x )( ∀ y )( ∃ z ) . . . ( ∀ w ) ∆ formula (cid:122) (cid:125)(cid:124) (cid:123) ( ∃ x (cid:48) < n ) . . . ( ∀ y (cid:48) < [ z (cid:48) + m ])( x = y + z ∨ . . . ∨ ( Prime ( z (cid:48) ) ∧ y (cid:48) | x ) ”The logical complexity of a given expression is governed by the block of unboundedquantifiers at its beginning when written in this way. Expressions are classifiedaccording to (1) the outermost quantifier (i.e., “ ∃ ” or “ ∀ ”) and (2) the number ofblocks of quantifiers of the same type. A formula that begins with a universal quan-tifier (“ ∀ ”) and is followed by n alternating blocks of quantifiers is a Π n formula; a The superscript 0 in Π refers to the fact that only first-order quantifiers are allowed; that is,quantifiers may only make assertions about objects in the structure referred to, rather than subsetsof the objects in a structure. If we allow expressions about sets of objects (e.g., “There exists a setof numbers such that every element except the first is double the previous element.”) then we havemoved beyond the so-called arithmetic formulas into second-order arithmetic. ∃ ”) and followed by n alternatingblocks of quantifiers is a Σ n -formula. Thus,“( ∀ x )( ∃ y )( x = 0 ∨ x = y + 1)”is a Π formula, as is “( ∀ x )( ∃ y )( ∃ z )( x = 0 ∨ x = z + y + 1)” . However, “( ∃ y )( y = 2)”is a Σ formula, and 2 = 1 ∗ x = 0are (possibly false) ∆ -formulas. (A formula is ∆ n if it is both Π n and Σ n . Notethat this agrees with our earlier definition of ∆ -formulas.) B.1.4 Π -formulas A particularly interesting class of formulas is those that live at the second level ofthe hierarchy, i.e., the Π -formulas. A large number of statements in elementarynumber theory take this form. For instance, Goldbach’s Conjecture may be statedas “( ∀ x )( ∃ y, z < x )(2 | x → ( Prime ( y ) ∧ Prime ( z ) ∧ x = y + z ))” . (I.e., “For all even x ∈ N , there exist primes y and z such that x = y + z . Here,“ Prime ( x )” is an abbreviation for the ∆ -formula “( ∀ y < x )( y | x → y = 1)” and“ y | x ” is an abbreviation for the ∆ -formula “( ∃ z < x )( y ∗ z = x )”.)Moreover, formalizations of consistency statements, such as “ PA does not provethat 0 = 1” or “ ZFC does not prove that 0 = 1” are likewise Π . The reason isstraightforward: informally, Π sentences express “for all” statements where whatcomes after the “for all” is something that for any given number n can be verifiedwith finite resources. The statement “ PA does not prove that 0 = 1” can beunderstood as “All valid proofs from the first order axioms of PA do not prove0 = 1” or, equivalently, “There is no valid proof in PA of the fact that 0 = 1”.Since proofs can be coded as numbers through the use of G¨odel numbering, andfor any given proof, it is possible to tell using finite resources whether it proves acontradiction such as “0 = 1”, these statements can be expressed using Π formulasin L PA .The question of whether a formula is Π is sometimes quite subtle. The Riemannhypothesis ( RH ), for instance, can be expressed in the form of a Π formula. Thisresult was first shown in [6], and a simpler expression was given in [17]. While fully74npacking its expression in L PA is outside the scope of this appendix, the Π formof RH given in [17] can be seen from the following paraphrase:“( ∀ n ) (cid:88) d | n d ≤ H n + exp( H n ) · log( H n ) ” , where H n = (cid:80) nj =1 1 j . B.1.5 Truth Formalizing the notion of truth is complicated. Tarski proved a theorem ( Tarski’stheorem ), that there can be no first order formula in L PA that completely captureswhat it means for a sentence in L PA to be true. That is, if we denote the G¨odelnumber of a formula ϕ by (cid:112) ϕ (cid:113) , there is no formula Tr for which it is always truethat Tr ( (cid:112) ϕ (cid:113) ) ↔ ϕ. In logical jargon, a formula that holds of the G¨odel numbers of sentences that aretrue and belong to some class Γ is called a truth predicate for Γ.The difficulty lurking behind Tarski’s Theorem lies in the fact that truth is anotion that is highly dependent on logical complexity. There is a truth predicatefor ∆ -formulas, but its expression is itself a ∆ -formula; likewise, we can constructtruth predicates for Π n - and Σ n -sentences which will themselves be Π n - and Σ n -sentences. For this reason, a general truth predicate would seem to have to be anelement of ∆ n for all n ∈ N , i.e., have infinite logical complexity.Since truth for ∆ -sentences can be defined in a ∆ way—that is, the truth of a∆ -sentence can be ascertained using only finite resources—and Π -sentences are ofthe form “( ∀ x, y, . . . , z ) F ( x, y, . . . , z )”, where “ F ( x, y, . . . , z )” is a ∆ -formula, wecan define truth for Π -sentences in a Π way. Definition 43 (∆ Truth) . Suppose “ F ” is a ∆ -sentence. Then “ F ” is equivalentto a Boolean combination of equalities and inequalities of polynomials in and .The formula “ F ” is true if and only if the Boolean combination is actually true in N . Definition 44 (Π Truth) . The Π -sentence “ ( ∀ x ∀ y . . . ∀ z ) F ( x, y, . . . , z ) ” is true ifand only if for all n , n , . . . , n k ∈ N , the ∆ -sentence “ F ( n , n , . . . , n k ) ” is true. Naively, the expression depends upon the logarithm and exponential functions, as well asrational numbers. Although we do not do so, it is possible to adequately “capture” these notions inplain arithmetic. The approach generally used in reverse mathematics—whose reliance on secondorder concepts is, in this case, superficial—is adequate to the task; see [22]. .2 Computability Theory Computability or recursion theory is the subdomain of logic concerned with thecomparative difficulty of various problems from the perspective of an idealized com-puter, such as a Turing machine with an infinitely long tape. It turns out thata large number of abstract models of computation coincide with one another andare captured by the notion of a recursive function. The celebrated Church’s The-sis is the meta-claim that all general notions a inherently finite computability areequivalent. The general reference for this section is [20]. B.2.1 Primitive recursion There are a large number of equivalent definitions of the primitive recursive func-tions. The most straightforward is to define a set of basic primitive recursive func-tions and then to define the primitive recursive functions as their closure undercertain operations. That is: • Initial primitive recursive functions: The following functions mapping N k to N for some k ≥ – Zero map: The function O ( x ) = 0. – Successor map: The function S ( x ) = x + 1. – Projection maps: The functions π k ( x , x , . . . , x k , . . . , x n ) = x k . • Composition: The primitive recursive functions are closed under composi-tion; that is, if f : N n → N is primitive recursive, and g , . . . , g n : N m → N areprimitive recursive, then the function h : N m → N given by h ( (cid:126)x ) = f ( g ( (cid:126)x ) , . . . , g n ( (cid:126)x ))is primitive recursive. • Primitive recursion: If g : N n → N is primitive recursive and h : N n +2 isprimitive recursive, then the unique function f : N n +1 → N satisfying f ( (cid:126)x, 0) = g ( (cid:126)x ) f ( (cid:126)x, y + 1) = h ( (cid:126)x, y, f ( (cid:126)x, y ))is primitive recursive.A wide variety of familiar “computable” procedures are primtive recursive, suchas addition, subtraction, powers, and logarithms. Even comparatively “expensive”operations, such as calculating the prime factorization of a given integer or solvingan arbitrary 3-SAT problem, are primitive recursive. This is because solving these As previously noted, we use the terms “computable” and “recursive” synonymously. bounded set of values the function can take. Informally, the set of primitiverecursive functions consists of all functions which can be implemented in a standardprogramming language using only FOR loops and not WHILE loops.What if one needs WHILE loops? The broader class of recursive functionsconsists of all problems which could in principle be solved by a computer like aTuring machine. Unlike primitive recursive functions, recursive functions can involve unbounded searches, and, indeed, it is not possible in general to determine whetheror not a recursive function is total (i.e., produces output for any given input).More precisely, the set of (partial) recursive functions is the smallest set offunctions containing the primitive recursive functions and closed under µ -recursion ;that is, given a recursive function g : N n +1 → N , the function f : N n → N given by f ( (cid:126)x ) = the least z ∈ N such that for all y ≤ z , g ( (cid:126)x, y ) is defined and g ( (cid:126)x, z ) = 0 , is also recursive. B.2.2 Computable real functions It is possible to extend the notion of a recursive function from the discrete domain N to a general continuous framework. For a more thorough introduction to the theoryof computable continuous functions, see [5].The primary obstacle to a theory of computable real functions is the observationthat uncountable metric spaces, such as [0 , ⊆ R , are inherently “incomputable”objects: there is no computer program that can enumerate all x ∈ [0 , , 1] does have a computable presentation , in the following sense: therationals are an inherently effective object, and we can imagine a recursive function n (cid:55)→ q n mapping n to the n -th rational in [0 , 1] in an effective enumeration. Then,the real numbers R can be “presented” as rapidly convergent sequences of the form (cid:104) x , x , x , . . . (cid:105) , where x i ∈ Q and | x i − x i +1 | < − ( n +1) .Since real numbers are given by rapidly converging sequences, computable func-tions on the real numbers should modify the sequences in a recursive way to producea new rapidly converging sequence. Intuitively, given a more and more accurate rep-resentation of the real number x —that is, the rapidly converging sequence of rationalnumbers (cid:104) x n (cid:105) —the function f should output more and more accurate informationabout the real number f ( x ) by producing a new rapidly converging sequence ofrational numbers.This process is most easily envisioned by means of a partial recursive function f : Q < N → Q mapping finite sequences of rational numbers to rational numbers. Strictly speaking, there is no such thing as a recursive map from N to Q as we have defined theterm “recursive.” However, one can “build” the rational numbers Q “inside” the natural numbers N in any of a variety of ways. For instance, pairs of integers can be “coded” by representing ( m, n )using the single integer ( m + n ) n + m . The rational number pq can be represented by the pair ( p, q ),etc. x represented by the sequence (cid:104) q i (0) , q i (1) , q i (2) , . . . (cid:105) , one can imagine successively feeding f the inputs (cid:126)q = (cid:104) q i (0) (cid:105) (cid:126)q = (cid:104) q i (0) , q i (1) (cid:105) (cid:126)q = (cid:104) q i (0) , q i (1) , q i (2) (cid:105) ...Let d ( i ) : N → N enumerate the q i for which f returns a value. Then, f ( x ) is thesequence (cid:104) f ( (cid:126)q d (0) ) , f ( (cid:126)q d (1) ) , f ( (cid:126)q d (2) ) , . . . (cid:105) . (Here, we assume that it does indeed hold that | f ( (cid:126)q d ( i ) ) − f ( (cid:126)q d ( i +1) ) | < − ( i +1) ;otherwise, the value of f at (cid:104) q i (0) , q i (1) , q i (2) , . . . (cid:105) is undefined.) For this reason, wecan speak of the computable continuous function f : R → R .It is clear that the basic notion is not changed if the rational approximations aremade with dyadic rationals. The computability assumption of f can be strength-ened. For example one can ask that f be primitive recursive. B.2.3 Modulus of continuity and approximation For the proof Theorem 1, we describe the notion of “computable real function” inan equivalent but slightly different way. We take them to have two parts:1. Modulus of continuity: The modulus of continuity is a recursive map d : N → N which calculates how much accuracy in the input is sufficient to geta desired accuracy in the output. In other words, an input accurate to 2 − d ( n ) ensures that the output can be specified to within 2 − n .2. Approximation: This is a recursive function g taking in a d ( n )-digit binarysequence s representing some q ∈ Q and producing an n -digit binary output g ( s ) which “approximates” the value of f at s .We illustrate this idea with the example of f ( x ) = exp( x ), x ∈ [0 , Modulus of continuity To define the modulus of continuity, as in the main text,we can avail ourselves of a Lipschitz constant. We illustrate the way we use theterms modulus of continuity and approximate in the main text with the example f ( x ) = e x . Since ddx exp( x ) = exp( x ), and max x ∈ [0 , exp( x ) < 3, it follows that | f ( x ) − f ( y ) | < | x − y | . Therefore, an input accurate to 2 − ( n +2) can be used to generate an output accurateto 2 − n places. Consequently, we set d ( n ) = n + 2.78 pproximation At the same time, we can generate good binary approximationsof exp( x ) in a primitive recursive way. Let s be an n -digit binary sequence repre-senting the dyadic rational k · − n for 0 ≤ k ≤ n . We simply set g ( s ) = n +3 (cid:88) j =0 ( k · − n ) j j ! n where [ x ] n denotes rounding x to its nearest length n binary approximation. ByTaylor’s Theorem, the m -th Taylor polynomial approximates exp( x ) on [0 , 1] towithin 6 /m !. Moreover, | x − [ x ] n | ≤ − ( n +1) . Therefore, it follows that the dyadicrational represented by g ( s ) is within 2 − n of exp( k · − n ).Putting together the modulus of continuity and approximation is sufficient torecover the computable real function exp( x ) exactly.This process is equivalent to the process outlined in section B.2.2. Suppose thereal number x is given by the rapidly converging sequence of rationals (cid:104) q i ( n ) (cid:105) . Then,to calculate the n -th rational in the representation of exp( x ), simply:1. Set m = d ( n ),2. Extract the binary sequence [ q i ( m ) ] m ,3. Calculate f ( s ),4. Output y = k · − m , where k · − m is the dyadic rational represented by thebinary sequence f ( s ).Note that y is a dyadic rational number. Since q i ( m ) approximates exp( x ) to within2 − ( n +2) , [ q i ( m ) ] n +2 approximates q i ( m ) to within 2 − ( n +2) , and the approximation f is accurate on n + 2-digit binary sequences to within 2 − ( n +2) , it follows that | y − exp( x ) | < − n , as desired. Computable C k functions Fix a k ∈ N . A function T : T → T is C k ifand only for all 0 ≤ i ≤ k (cid:48) ≤ k, j ∈ { , } , the partial derivatives ∂ k (cid:48) ∂ i x∂ k (cid:48)− i y ( T j )is uniformly continuous. For each k (cid:48) , j, i we can apply the notions of modulus ofcontinuity and approximation to ∂ k (cid:48) ∂ i x∂ k (cid:48)− i y ( T j ) individually and ask the ensemble ofpartial derivatives as 0 ≤ i ≤ k (cid:48) ≤ k, j ∈ { , } is computable. This gives thedefinition of a computable C k function. Similarly for k = ∞ we ask that for all0 ≤ i ≤ k (cid:48) ∈ N , j ∈ { , } , ∂ k (cid:48) ∂ i x∂ k (cid:48)− i y ( T j ) is uniformly continuous and that thereis a single algorithm that computes the modulus of continuity and approximationuniformly in k (cid:48) .This generalizes easily to arbitrary smooth manifolds, adding complexity withoutcontent. 79 Ergodic Theory Background In this appendix, we briefly explain and define important notions in ergodic theoryrelevant to Sections 2 to 3. C.1 Why Z ? Why T ? Why C ∞ ? The short answer is that we want to work in the simplest, best behaved and mostclassical context.Physical systems are often modeled by ordinary differential equations on a smoothcompact manifold M . Solutions are formalized as dynamical systems: ϕ : R × M → M such that ϕ ( s, ϕ ( t, x )) = ϕ ( s + t, x ) and ϕ ( s, · ) : M → M is measure preserving.Doing repeated experiments in a physical realization of such a system—say tomeasure a constant of interest such as the average value of an L function on M —is viewed as measuring ϕ ( t , x ) , ϕ ( t + t , x ) , . . . ϕ (( N − t , x ) and averaging: N (cid:80) i f ( ϕ ( i ∗ t , x )). Provided that the system is sufficiently mixing (“ergodic”),the Ergodic Theorem implies that for almost every x the averages along trajectoriesconverge to the integral of f over N .Thus empirical experiments are construed as sampling along portions of a Z -action given by: ψ ( n, x ) = ϕ ( nt , x ) . The manifold is required to be compact to avoid wild behavior and asked to be ofthe smallest possible dimension. Dimension one is impossible because there are veryfew conjugacy classes of measure preserving diffeomorphisms on one dimensionalmanifolds. On the unit circle there are exactly two.Thus we move to two dimensional compact manifolds. The most convenientchoice is T , the two torus.As k increases, the behavior of C k diffeomorphisms becomes more regular–thebehavior of C -diffeomorphisms can be quite wild. Thus the theorem involves C ∞ -diffeomorphisms because it illustrates that the basic issue is not how wild the dif-feomorphism is.It could be argued that the tamest situation of all involves real analytic trans-formations of the 2-torus. The results in this paper can be extended to real-analyticmaps using the work of Banerjee and Kunde [3]. In Summary We are proving that the question of forward vs. backward timeencodes some of the most complex problems in mathematics. This claim is madestronger by taking the simples possible context: time is given by a Z -action, T isthe simplest, most concrete manifold possible, and the diffeomorphisms in questionare the most regular possible. 80 .2 Basics Unless otherwise stated, all definitions are modulo a null set. That is, A = B isshorthand for µ [ A (cid:52) B ] = 0, f = g is shorthand for µ [ { x : f ( x ) (cid:54) = g ( x ) } ] = 0, andso on. Put differently, all statements in this appendix contain a tacit “a.e.” Definition 45 (Measurable Dynamical Systems) . A measurable measure preservingsystem is a quadruple ( X, F , µ, T ) where: • X is a set, • F is a σ -algebra of subsets of X , • µ is a measure on F such that µ [ X ] = 1 , and • T : X → X is an invertible, F -measurable, µ -preserving transformation on X , i.e., µ [ A ] = µ [ T − ( A )] .An ergodic system (or, more specificially, an ergodic transformation ) is a measur-able dynamical system in which T satisfies the additional hypothesis that if T − ( A ) = A , then A = X or A = ∅ a.e. One way of understanding this definition is that ergodic systems are the atomicunits of measurable dynamical systems; measurable dynamical systems can be bro-ken down into ergodic pieces, but no further. This intuition is formalized by theergodic decomoposition theorem; see Theorem 3.22 in [15].Measurable dynamical systems can be studied from several perspectives. Inaddition to their straightforward pointwise actions, they can also be viewed fromthe perspective of functional analysis. Definition 46 (Koopman Operator) . Let T : X → X be a measure-preservingtransformation. Then T induces an operator U T : L p ( X ) → L p ( X ) in the followingway: f (cid:55)→ f ◦ T. The operator U T is known as the Koopman operator . The following theorem is a fundamental tool in ergodic theory. Theorem 47 (Pointwise Ergodic Theorem) . Let ( X, F , µ, T ) be a measure-preservingdynamical system. Then, for any f ∈ L ( X ) and a.e. x ∈ X , lim n →∞ n n − (cid:88) k =0 f ( T k x ) , converges a.e. to an L ( X ) function ¯ f , which is invariant under the action of T and (cid:82) X ¯ f dµ = (cid:82) X f dµ . T is ergodic, the only invariant functions are constant a.e. on X . Therefore, inthe ergodic case, for a.e. x ∈ X ,lim n →∞ n n − (cid:88) k =0 f ( T k x ) = (cid:90) X f dµ C.3 Symbolic systems Symbolic systems, some of which are examined in detail in Section 2 and Subsec-tion 3.1, represent a large and well-studied class of measurable dynamical systems.What follows in this subsection collects and further elaborates on the backgroundgiven in Section 2. Definition 48 (Symbolic System) . Let Σ be some finite or countable alphabet. Then Σ Z is the collection of all bi-infinite words in Σ , i.e., functions s : Z → Σ . Let Sh represent the shift operator, i.e., for all n ∈ Z , Sh ( s )( n ) = s ( n + 1) . Let B be a shift-invariant σ -algebra of subsets of Σ Z . Lastly, let µ be a shift-invariantprobability measure measure on B . Then we call the quadruple (Σ Z , B , µ, Sh ) a symbolic system . A common way of constructing symbolic systems is by combining finite wordsinto infinite words according to some specified pattern. For finite words, we makeuse of the following notation and conventions: • The length of a word u is | u | ; • The concatenation of the words u and w is written u · w , or, where clear fromcontext, uw ; • The repeated concatenation of a word with itself is denoted as follows: w n = n -times (cid:122) (cid:125)(cid:124) (cid:123) w · w · · · w ;and, • All finite words are assumed to be zero-indexed, i.e., if | w | = n , then w (0) isthe first letter in w and w ( n ) is undefined. • For a finite or infinite word s , s (cid:22) [ n, m ) for n, m ∈ Z is the finite subwordbeginning at n and having length m − n .82 efinition 49 (Unique Readability) . A collection of words W is uniquely readable when for any u, v, w ∈ W , if u · v = p · w · p (cid:48) , then either p or p (cid:48) is the empty word. Definition 50 (Construction Sequence) . Let Σ be a finite or countable alphabet. Wecall a squence of sets of finite words (cid:104)W n : n ∈ N (cid:105) in the alphabet Σ a constructionsequence if:1. All the words in W n have the same length q n ;2. Each w ∈ W n +1 contains each v ∈ W n as a subword;3. For all n > , if w ∈ W n +1 , then there is a unique parsing of w into segmentsin terms of w , . . . , w l ∈ W n , u , . . . , u l +1 / ∈ W , such that w = u · w · u · · · w l · u l +1 . (34) 4. There is a summable sequence (cid:104) (cid:15) n : n ∈ N (cid:105) such that if u , . . . , u l +1 are as inEquation 34, then (cid:80) l +1 i =0 | u i | q n +1 = (cid:15) n +1 . (35) In Equation 34, we call the words u i spacers .A construction sequence gives rise to a symbolic dynamical system in a naturalway. Define the set K to be { s ∈ Σ Z : ( ∀ n, m )( ∃ k, N ) s (cid:22) [ n, m ) = w (cid:22) [ n + k, m + k ) for some w ∈ W N } . That is, K consists of all bi-infinite words with the property that every finite subwordis itself a subword of some word w ∈ W N for some N ∈ N .Let B be the Borel sets and let ν be some Sh -invariant measure on B such that ν [ S ] = 1 . Then the system ( S, B , ν, Sh ) is the symbolic system corresponding to theconstruction sequence (cid:104)W n : n ∈ N (cid:105) . Remark 51. While unique readability and construction sequences are fundamentalfor the systems constructed in this paper they are not a property of typical symbolicsystems. Suppose that v ∈ W n and w ∈ W n +1 . Let r ( v, w ) be the number of occurrences of v in w . Definition 52 (Uniform Construction Sequence) . We say that a construction se-quence is uniform if there is a summable sequence (cid:104) (cid:15) n : n ∈ N (cid:105) and a sequence (cid:104) d n : n ∈ N (cid:105) in (0 , of densities such that for all w ∈ W n +1 and v ∈ W n , (cid:12)(cid:12)(cid:12)(cid:12) r ( v, w ) q n +1 /q n − d n (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) n . hat is, each v ∈ W n occurs in each w ∈ W n +1 with approximately the same fre-quency as all of the others.If r ( v, w ) is actually constant over v ∈ W n and w ∈ W n +1 , then we say that (cid:104)W n (cid:105) is strongly uniform . Definition 53 (Cylinder Sets) . Let w be a finite word in Σ . Then the cylinder set associated to w , also written (cid:104) w (cid:105) , is given by (cid:104) w (cid:105) = { s ∈ Σ Z : s (cid:22) [0 , | s | ) = w } . These notions are useful largely for the following lemma which is proved in [13],which allows us to further restrict the set K to especially well-behaved bi-infinitewords s . Lemma 54. Let (Σ Z , B , ν, Sh ) be a symbolic system arising from the constructionsequence (cid:104)W n (cid:105) . Then:1. The set K is the smallest closed shift-invariant subset of Σ Z with non-emptyintersection with every basic open set (cid:104) w (cid:105) for w ∈ W n for some n .2. Suppose (cid:104)W n (cid:105) is uniform. Then, let S be the collection of s ∈ Σ Z such thatthere are increasing integer sequences (cid:104) a n : n ∈ N (cid:105) and (cid:104) b n : n ∈ N (cid:105) suchthat a n , b n → ∞ as n → ∞ and s (cid:22) [ − a n , b n ) ∈ W n . Then there is a uniquenon-atomic shift-invariant measure ν concentrating on S , and this ν is ergodic. In Lemma 54, the fact that ν exists and is unique is proved in Lemma 11 of [13].For strongly uniform systems it is a direct consequence of the ergodic theorem thatfor a generic x ν ( (cid:104) v (cid:105) ) = lim n →∞ |{ Occurences of w in x (cid:22) [ − a n , b n ) }| n which, because (cid:104)W n (cid:105) is strongly uniform, is fixed independently of x . C.4 Odometers Odometers are an important class of transformations. They give a symbolic re-alization of the Kronecker factors of the odometer-based transformations built inSection 2 using the Substitution Lemma. Definition 55 (Odometers) . Let (cid:104) k n : n ∈ N (cid:105) , k n > be a sequence of naturalnumbers. Then the group O = ∞ (cid:89) i =0 Z /k n Z is the (cid:104) k n (cid:105) -adic integers. By giving each factor of the form Z /k n Z a uniform mea-sure, we can form the product measure µ . (This is also the Haar measure on O .) efine the transformation O which acts by (cid:104) k n (cid:105) -adic addition by one. That is,the transformation is given by addition by one with a carry to the right: T : (cid:104) x , x , x , . . . (cid:105) (cid:55)→ (cid:104) x + 1 , x , x , . . . (cid:105) , unless x = k − , in which case T : (cid:104) x , x , x , . . . (cid:105) (cid:55)→ (cid:104) , x + 1 , x , . . . (cid:105) , unless both x = k − and x = k − , in which case T : (cid:104) x , x , x , . . . (cid:105) (cid:55)→ (cid:104) , , x + 1 , . . . (cid:105) , and so on.A measure preserving system given in this way is called an odometer . A useful property of the odometers is that the eigenvalues of their associatedKoopman operators are easily characterized. Theorem 56. Let q n = k · k · · · k n − , and let A n ⊆ O be the set of points whosefirst n coordinates are zero. Define R n = q n − (cid:88) j =0 ( e πi/q n ) j · χ O j ( A n ) , where we denote the result of applying the transformation O j − times to A n by O j ( A n ) . Then: • The function R n is an eigenvector of U O with eigenvalue e πi/q n , • The function R q n +1 n +1 is equal to R n , and • The collection {R kn : 0 ≤ k ≤ q n , n ∈ N } form a basis of L ( O ) . Odometer systems are ergodic and canonically isomorphic to their inverses viathe map x (cid:55)→ − x .We have the following immediate corollary, which gives a simple sufficient condi-tion for two parameter sequences (cid:104) k n (cid:105) and (cid:104) k n (cid:105) define distinct (i.e., non-isomorphic)odometers. Corollary 57. Let O and O be odometers with associated parameter sequences (cid:104) k n (cid:105) n and (cid:104) k n (cid:105) n . Let p = { p ∈ N : p is prime and for some m ∈ N , p | k m } , and p = { p ∈ N : p is prime and for some m ∈ N , p | k m } . Then, if p (cid:54) = p , O (cid:54)∼ = O .Proof. The Koopman operators U O and U O are invariants of the respective odome-ters. By Theorem 56, the eigenvalues of these operators are roots of unity, and theprime divisors of these orders are p and p , respectively. (cid:97) .5 Factors, joinings, and conjugacies Like most objects, measurable dynamical systems interact with one another viacertain kinds of morphisms. These morphisms must respect the action of the trans-formations, but only need be defined a.e. Definition 58 (Factors) . Let ( X, B ( X ) , µ, T ) and ( Y, B ( X ) , ν, S ) be ergodic systems.A map ϕ : X → Y is a factor if and only if ϕ ∗ µ , the pushforward measure, equals ν , and the following diagram commutes for almost all x ∈ X : X XY Y Tϕ ϕS A map ϕ meeting these requirements is a factor map and ( Y, B ( X ) , ν, S ) is said tobe a factor of ( X, B ( X ) , µ, T ) . (More briefly, S is a factor of T .) Definition 59 (Isomorphism) . Let X = ( X, B ( X ) , µ, T ) and Y = ( Y, B ( X ) , ν, S ) be ergodic systems. Then X is measure isomorphic to Y if and only if there is aninvertible factor map ϕ : X → Y . We write S ∼ = T .Informally we say “ S is isomorphic” or “ S is congruent” to T . It is easy toverify that if X = Y , then S ∼ = T if and only if S and T are conjugate in the groupof measure preserving transformations of X . For this reason we use congruent as asynonym of isomorphic . We now consider generalizations of factor maps, joinings. Definition 60 (Joining) . A joining of two ergodic systems X = ( X, B ( X ) , µ, T ) and Y = ( Y, B ( Y ) , ν, S ) is a T × S -invariant measure η on ( X × Y, B ( X ) ⊗ B ( Y )) suchthat: • η ( A × Y ) = µ ( A ) , and • η ( X × B ) = µ ( B ) . Let η be a joining of X and Y . If X (cid:48) , Y (cid:48) are factors of X , Y , respectively, thenthe measure algebras associated with X (cid:48) and Y (cid:48) can be viewed as subalgebras of themeasure algebras of X and Y . Hence η induces an invariant measure on X (cid:48) ⊗ Y (cid:48) andhence a joining η (cid:48) of X (cid:48) and Y (cid:48) . Definition 61 (Graph Joinings) . If π : X → Y is a factor map, then viewed asa subset of X × Y , π is T × S -invariant and can be canonically identified with ajoining J by setting: J ( A ) = µ ( { x : ( x, π ( x )) ∈ A } ) . Joinings coming from factor maps are called graph joinings . An invertible graphjoining is a graph joining coming from an isomorphism. Definition 62 (Disintegration) . Let ϕ : X → Y be a factor map, and, for y ∈ Y , A ⊆ X , let A y be the fiber of A over y . Then there exists a family of measures { ν y : y ∈ Y } concentrating on respective fibers X y , such that:1. Each ν y is a standard probability measure on X y ;2. For A ∈ B ( X ) , r ∈ [0 , , and (cid:15) > , the set { y ∈ Y : A y is ν y -measurable and | ν y ( A y ) − r | < (cid:15) } is ν -measurable; and3. For all A ∈ B ( X ) , µ ( A ) = (cid:82) Y ν y ( A y ) dν .The family { ν y : y ∈ Y } is the disintegration of ν over ϕ . Definition 63 (Relatively Independent Joinings) . Let X = ( X, B , µ, T ) , Y = ( Y, C , ν, S ) ,and Z = ( Z, D , ρ, R ) be ergodic systems such that X YZ ϕ ϕ and { µ z } , { ν z } be the respective disintegrations. Then define η , the relatively inde-pendent joining of X and Y over Z by η ( A × B ) = (cid:90) Z µ z ( A z ) ν z ( B z ) dρ ( z ) which concentrates on { ( x, y ) : ϕ ( x ) = ϕ ( y ) } . D Diffeomorphisms This appendix gives a very brief description of the role of diffeomorphisms play inthis paper, as opposed to abstract measure-preserving transformations.87 .1 Diffeomorphisms of the torus Definition 64 (The two-torus) . We define the two-torus T to be the product of theunit interval with itself with opposite edges identified: define an equivalence relationon [0 , × [0 , by setting (0 , y ) ∼ (1 , y ) for all y and ( x, ∼ ( x, for all x . Weview T as the unit square modulo the equivalence relation, i.e., T = [0 , × [0 , / ∼ . For this reason, to ensure that a continous map T : [0 , × [0 , → [0 , × [0 , torus , it suffices to ensure that1. for all x, T ( x, 0) = T ( x, 1) and for y, T (0 , y ) = T (1 , y ),2. there is an R neighborhood U of [0 , × [0 , 1] such that T can be extendedsmoothly to a diffeomorphism T ∗ : U → U such that T ∗ ( x, y ) = T ([ x ] , [ y ] ).The set of C ∞ -maps on a compact manifold have a natural topology in whichtwo maps are close if the transformations themselves are close, their differentials areclose, and so on. More precisely, if M is a C k -smooth compact finite-dimensionalmanifold and µ is a standard measure on M determined by a smooth volume el-ement, then, for each k there is a Polish topology on the k -times differentiablehomeomorphisms of M . The C ∞ -topology is the coarsest common refinement ofthese topologies.This topology is induced by a very explicit metric. Definition 65 (The d ∞ metric) . Let S and T be smooth maps from T to T . Then,set the distance between S and T , d ∞ ( S, T ) as follows: d ∞ ( S, T ) = ∞ (cid:88) k =0 − k · max x ∈ T || D k S ( x ) − D k T ( x ) || x ∈ T || D k S ( x ) − D k T ( x ) || . (36)Here D k f denotes the k -th differential of f . Note that this measure is effective in the sense that to determine the distance between S and T to an accuracy of 2 − k ,it is only necessary to examine the first k differentials of S and T .The two-torus carries a natural Lebesgue measure and the diffeomorphisms weconsider will preserve the Lebesgue measure. However the conjugacy relation be-tween diffeomorphisms does not require smoothness: two diffeomorphisms S, T areconjugate if and only if there is a (not necessarily smooth) invertible measure pre-serving transformation θ such that: θ ◦ T = S ◦ θ. .2 Smooth permutations In our construction of diffeomorphisms of T , it was required to find smooth approx-imations to arbitrary permutations of a rectangular partition of a larger rectangle.This is discussed in Theorem 41. This is done by composing C ∞ maps that swapmost of the interior of adjacent squares. These maps emulate the transpositionsused to construct an arbitrary permutation. (See [13]). For completeness we ex-hibit concrete proofs to illustrate that the swap-diffeomorphisms can be taken to beprimitive recursive.As throughout the paper we say smooth to mean C ∞ . When the domain hasboundary we mean C ∞ in the interior and continuous at the boundary. One ingre-dient is the following example: Example 66. (Bump function) Define g ( x ) = (cid:40) e − x , x > , x ≤ and set f ( x ) = g ( x ) g ( x ) + g (1 − x ) . One verifies directly that f ∈ C ∞ , f (cid:22) ( −∞ , 0] = 0 and f (cid:22) [1 , ∞ ) = 1 and f mapsthe unit interval to the unit interval.By rescaling and translating the input to f and possibly replacing it by − f , forall α < β ∈ R , a ∈ { , } one can create total C ∞ functions that map [ α, β ] to [0 , ,are constantly a on ( −∞ , α ] and − a on [ β, ∞ ) . Here is an explicit version of Lemma 36 in [13]. The author is grateful to A.Gorodetski for providing the complex analysis background for this argument. Areference for the results cited is [7]. A guide to a Matlab software package forcomputing the relevant mappings claimed is available at . Lemma 67. Let A = [0 , × [ − , and (cid:15) > . Then there is an area preservingmap f : A → A that is C ∞ on the interior of A , is the identity on a neighborhoodof the boundary of A and there is an open set O ⊆ A such that1. O is symmetric about the x axis,2. λ ( O ) > − (cid:15) ,3. For each x on the boundary of O the distance from x to the boundary of A isless than (cid:15) .4. f maps the top half of O to the bottom half of O . Let D be the closed disk of area 2 centered at the origin viewed as a subset of C . By the Schwarz-Christoffel theorem there is an analytic bijection θ from theupper half disk, D ∩ { Im z > } , to [0 , × [0 , 1] that is analytic on the interior andcontinuous on the boundary. Moreover it sends the intersection of the disk with the x -axis to [0 , × { } .By the Schwartz Reflection Principle, if θ is extended symmetrically about the x -axis the result (which we also call θ ) is still analytic. (So the map θ satisfies theidentity θ (¯ z ) = θ ( z ).) Note: For uncomplicated regions such as the half-disk and a square, there areexplicit integral formulas for the Schwarz-Christoffel mapping. These are given in[7]. Using a slight variation on Moser’s ‘Lemma 2’ in [19], this map can be composedwith another map so that the result θ is measure preserving, C ∞ and still symmetricabout the horizontal line.Since the proof of Moser’s lemma in [19] is a little bit confusing, the following isa painfully explicit proof of a slight variation of a very special case. Lemma 68. (Moser prime) Suppose that θ : D → A is a bijection that is analyticon the interior of D. Then there is a bijection u : A → A that is C ∞ in the interiorof A and preserves the boundary of A such that θ = u ◦ θ is Lebesgue measurepreserving and takes the top half of A to the top half of A . (cid:96) Let µ be the measure on A given by µ ( A ) = λ ( θ − ( A )) . Then µ is absolutely continuous with respect to Lebesgue measure on A . Let h bedensity associated with µ . Then h is continuous and analytic on the interior of A .We first show that it suffices to find a u : A → A such that the Jacobian of u is h .Let θ = u ◦ θ and X ⊆ A . Then: λ ( X ) = (cid:90) X dλ = (cid:90) u − ( X ) (1 ◦ u ) det ( D ( u )) dλ = (cid:90) u − ( X ) h dλ = λ (( θ ◦ u ) − ( X ))= λ ( θ − ( X )) . Let u = ( u , u ) and suppose that u is a function of x and u is a functionof ( x , x ). Then det ( D ( u )) = du dx du dx . We rewrite h = h ( x ) h ( x , x ) and find90unctions u , u so that du dx = h ( x ) (37) du dx = h ( x , x ) . (38)Set h ( x ) = (cid:90) − h ( x , t ) dth ( x , x ) = h ( x , x ) h ( x )Then equations 37 and 38 give two equations that can be solved effectively byordinary integration and yield smooth solutions. u ( x ) = (cid:90) x h ( x )= (cid:90) x (cid:90) − h ( x , t ) dt and u ( x , x ) = (cid:90) x h ( x , x ) dx = (cid:82) x h ( x , x ) dx (cid:82) − h ( x , t ) dt . Clearly u (0) = 0 and u (1) = 1 and u ( x , 0) = 0 , u ( x , 1) = 1, so u preserves theboundary.By the symmetry of θ , for each x i , (cid:90) − h ( x , t ) dt = (cid:18) (cid:19) (cid:90) − h ( x , t ) dt. It follows immediately that u ( x , x ) ≥ x ≥ 0. Hence u takes the top half of A to the top half of A . (In fact it is easy to verify that u is symmetric about the x -axis.)It is also easy to verify that u is one to one. If x (cid:54) = y then u ( x ) (cid:54) = u ( y ). If u ( x ) = u ( y ) but x (cid:54) = y then u ( x , x ) (cid:54) = u ( y , y ). Since u is continuous andtakes the faces of the A to faces of A it follows that u is a surjection. (cid:97) We now finish the proof of Lemma 67. Let R be the radius of the disk D ofarea 2. By the uniform continuity of θ we can choose an γ > R − γ is sent to a point within (cid:15) of the boundary of A . Without loss of generality, the area of the disk of radius R − γ is bigger than2 − (cid:15) . As in Lemma 36 of [13] we consider a C ∞ function F : [0 , R ] → [0 , π ] suchthat F (cid:22) [0 , R − γ ] is identically equal to π , and F is identically equal to zero ina neighborhood of R . Define a C ∞ , area preserving map ϕ : D → D in polarcoordinates by ϕ ( r, θ ) = ( r, θ + F ( r ))Then ϕ rotates the upper half disk of radius R − γ to the lower half disk of radius R − γ . Such a function is given explicitly in Lemma 66.It is now routine to check that the map f = θ ◦ ϕ ◦ θ − satisfies the conclusionsof Lemma 67. (cid:97) References [1] D. V. Anosov and A. B. Katok. New examples in smooth ergodic theory.Ergodic diffeomorphisms. Trudy Moskov. Mat. Obˇsˇc. , 23:3–36, 1970.[2] Hirotada Anzai. On an example of a measure preserving transformation whichis not conjugate to its inverse. Proc. Japan Acad. , 27:517–522, 1951.[3] S. Banerjee and P. Kunde. Real-analytic abc constructions on the torus. ErgodicTheory and Dynamical Systems , 39(10):2643–2688, 2019.[4] Shilpak Banerjee. Non-standard real-analytic realizations of some rotations ofthe circle. Ergodic Theory Dynam. Systems , 37(5):1369–1386, 2017.[5] Vasco Brattka, Peter Hertling, and Klaus Weihrauch. 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