Combinatorial invariants of metric filtrations and automorphisms; the universal adic graph
CCombinatorial invariants of metric filtrations and automorphisms; theuniversal adic graph ∗ A. M. Vershik † P. B. Zatitskiy ‡ Contents Z and classification of combinatorially definite filtrations . . . 6 Abstract
We suggest a combinatorial classification of metric filtrations in measure spaces; a completeinvariant of such a filtration is its combinatorial scheme, a measure on the space of hierarchies ofthe group Z . In turn, the notion of combinatorial scheme is a source of new metric invariants ofautomorphisms approximated via basic filtrations. We construct a universal graph endowed withan adic structure such that every automorphism can be realized in its path space. ∗ Supported by the RSF grant 17-71-20153. † St. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg State University, and Institute forInformation Transmission Problems. E-mail: [email protected] . ‡ St. Petersburg State University and St. Petersburg Department of Steklov Institute of Mathematics. E-mail: [email protected] . a r X i v : . [ m a t h . D S ] D ec Introduction
This paper deals with applications of the theory of metric filtrations (see [16] and the referencestherein) to uniform approximation of automorphisms of measure spaces and the analysis of adictransformations in path spaces of graphs. Conceptually, it is closely related to the first author’swork on dyadic and homogeneous sequences of measurable partitions (= filtrations), standardness,the “scale” metric invariant, etc. (see [8, 9, 10, 12, 11, 15, 17]). Essentially, we pass from homoge-neous filtrations to arbitrary ones, pose a number of problems on so-called combinatorially definite (standard) filtrations, and relate them to properties of automorphisms being approximated. On theother hand, the main example of filtrations is provided by so-called tail filtrations in path spaces ofgraded graphs, or, equivalently, spaces of realizations of Markov chains, so we arrive at a realizationof automorphisms as adic transformations (see [13, 14]).It is characteristic of numerous classification problems in ergodic theory that the most importantobjects (automorphisms, group actions) have no nontrivial finite invariants, i.e., metric invariantsarising from finite approximations or finite projections. This means that classification problems areof purely asymptotic nature. An illustration of this point is, for example, the classical Rokhlin’slemma, which says that every aperiodic automorphism (more exactly, every free action of thegroup Z with invariant measure) can be approximated with any accuracy in all reasonable metricsby periodic automorphisms. Hence, to obtain nontrivial invariants of automorphisms, one shouldconsider infinite sequences of periodic approximations and their invariants.There are two theories of infinite approximations by periodic transformations: the theory ofweak approximations, in the weak (operator) topology, successfully developed in the 1960s–1970sby A. Katok, A. Stepin, and others (see [6, 4, 7]), and the theory of uniform approximations,in the uniform metric, initiated by the first author in the 1960s simultaneously with the theoryof filtrations, i.e., decreasing sequences of σ -algebras or measurable partitions, see the referencesabove. The theory of uniform approximations and orbit theory were the main applications of thetheory of filtrations. Another important area of application for the theory of filtrations, whichwe do not touch upon in this paper, is the theory of stationary filtrations arising as decreasingsequences of σ -algebras of “pasts” of stationary random processes, or, which is the same, sequencesof preimages of the full σ -algebra under powers of faithful endomorphisms.In this paper, we explain that a monotone sequence of uniform approximations of an auto-morphism in a measure space determines, in a natural way, a filtration whose partitions are orbitpartitions of periodic automorphisms. This filtration is special in the sense that it is endowedwith an order and is semihomogeneous; in other words, it inherits two approximation structures:a coherent ordering of points in almost all elements of all partitions (a linear order in the group Z )and semihomogeneity, i.e., the uniformness of almost all conditional measures, which follows fromthe invariance of the measure under the automorphism. In terms of the theory of graded graphs,this means that the graph is endowed with a structure of a linear order on the edges entering eachvertex (an “adic structure”), and that the measure on the path space is central, i.e., the conditionalmeasure on initial segments of paths is uniform. Moreover, a semihomogeneous filtration endowedwith such an order uniquely determines the corresponding automorphism (without order, one can-not recover the automorphism from the filtration up to isomorphism). From this viewpoint, thefiltration approach and uniform approximation are related to the problem of metric isomorphismmore closely than weak approximation. The study and construction of invariants of automorphisms nd groups of automorphisms is preceded by the study of invariants of filtrations. All metric invariants of filtrations fall into two classes: combinatorial (finite) invariants andtransfinite ones . Combinatorial invariants are invariants of all finite fragments of filtrations, i.e.,invariants of periodic approximations; they are described below and represent some measures onthe space of hierarchies on the group Z . The prospect of obtaining an efficient combinatorialclassification of filtrations described below was observed in [16], but the fact that this classificationproblem has indeed turned out to be tame gives hope for further classifications.Transfinite invariants, whose existence is not obvious, are not combinatorial; their study requiresconsidering deeper properties of filtrations, related to the notion of standardness or combinatorialdefiniteness. Standard, or combinatorially definite, filtrations are filtrations that are uniquelydetermined up to metric isomorphism by the combinatorial invariants. For example, a dyadicstandard filtration is a Bernoulli filtration, it is combinatorially definite and can be recovered fromits one-dimensional distribution.By the lacunary theorem (see [8, 16] and Section 4), every filtration contains a “thinning” thatis already a combinatorially definite filtration; thus, every automorphism becomes combinatoriallydefinite with respect to some “thinning.” So, we obtain a chain of invariants also for filtrationsthat are not combinatorially definite, and for general automorphisms.The adic dynamics (see [13, 14]), i.e., a special transformation of the path space of a gradedgraph, has already given many new nontrivial examples of dynamical systems. For instance, thePascal automorphism (whose spectrum is still unknown) is combinatorially definite in the sense ofthis paper. In general, adic transformations are of great interest both from theoretical and practicalpoint of view. Here we suggest constructions of universal graphs on which every automorphismcan be realized as an adic shift. The study of specific automorphisms has already begun (see thesurvey [1], and also [2]), but, according to the conclusion of the paper [16] (see also below), theproblem of classification of combinatorially definite filtrations is tame, i.e., there is a manageablespace of classes, or orbits, or complete metric invariants of such filtrations (see the definition ofcombinatorial schemes of hierarchies below). This gives a chance to obtain a reasonable combina-torial classification of measure-preserving automorphisms. One may hope that such an approachis also possible for actions of other (amenable) groups, primarily for locally finite groups, suchas (cid:80) Z p , and for the lattices Z d . Recall, in a form suitable for our purposes, the well-known Rokhlin’s lemma on approximation ofa measure-preserving aperiodic automorphism by periodic automorphisms in the uniform metric.The uniform metric on the space of transformations of a space X preserving a measure µ is definedas follows: given transformations T and ˜ T , ρ ( T, ˜ T ) ≡ µ { x ∈ X : T x (cid:54) = ˜
T x } . Lemma 1 (Rokhlin’s lemma) . For every ε > , every positive integer n ∈ N , and every measure-preserving aperiodic automorphism T in a Lebesgue space ( X, µ ) there exists a periodic automor-phism T n with period n such that ρ ( T, T n ) < n + ε. n what follows, it is useful to drop the condition that T n has period n almost everywhereand assume that the periods p n ( x ) of T n can be different at different points x ∈ X but uniformlybounded on a subset of full measure. This weaker assertion is even slightly easier to prove thanthe classical Rokhlin’s lemma. Denote by ORB( T n ) the measurable partition of the space ( X, µ )into the orbits of T n ; we may assume that almost all elements of this partition are finite sets ofthe form { x, T x, . . . , T k x } endowed with the uniform conditional measure and a linear order suchthat the resulting ordered space is isomorphic to an arbitrary interval of the line (we set y < T y for every y ).Greater freedom in choosing approximations is needed to construct a coherent sequence ofapproximations satisfying the following properties: ρ ( T n , T ) →
0, and the partitions ORB( T n )become coarser, i.e., almost every orbit of T n +1 consists of several orbits of T n . Besides, for every n the quotient of ORB( T n +1 ) by ORB( T n ) is endowed with a linear order. Let us introduce anabstract definition for the resulting structure. Definition 1.
A filtration in a Lebesgue space (
X, µ ) with continuous measure is a decreasingsequence of measurable partitions Ξ = { ξ n } n ≥ with ξ = ε being the partition into singletons.A filtration is said to be ergodic if the measurable intersection (cid:86) n ξ n is the trivial partition, whichis usually denoted by ν .We introduce the following special properties of filtrations. A filtration is said to • be locally finite if for every n almost all elements of ξ n are finite sets, and the number ofdifferent types of conditional measures on elements of ξ n is finite (depending on n ); • be semihomogeneous if the conditional measures on the elements of ξ n are uniform for all n ; • induce an order if every element of the quotient partition ξ n +1 /ξ n is endowed with a mea-surable linear order (measurability means that the set of all points with a given number in theelements of ξ n +1 /ξ n is measurable); these orders induce a coherent order on the elements ofthe partitions ξ n , and hence on the classes of the limiting partition (cid:84) n ξ n (which is in generalnot measurable); we assume that the order type is Z for almost all classes.Filtrations satisfying all these properties are said to be basic . Definition 2.
Let T be an aperiodic automorphism of a Lebesgue space ( X, µ ). We say thata basic filtration Ξ = { ξ n } of ( X, µ ) is basic for T if the corresponding order is induced by T andthe limit of ξ n is the partition into the orbits of T mod 0.Of course, for every aperiodic automorphism T there are many basic filtrations. If T is ergodic,then its basic filtrations are also ergodic. Given a basic filtration for T , one can construct a coherentsequence of automorphisms T n approximating T in the sense described above. So, the language ofbasic filtrations and that of coherent sequences of approximating automorhisms are equivalent. Let A be an arbitrary finite set and { η i } ni =0 be an ordered finite filtration on this set, the lastpartition η n being trivial (consisting of a single nonempty class). We construct an ordered graded ree corresponding to this finite filtration as follows. The vertices of level i in this tree correspondto the elements of η i . A vertex of level i + 1 is joined by an edge with a vertex of level i if thecorresponding elements of partitions are nested. The n th level contains a single vertex, while thevertices of level 0 are the elements of A . The set A is endowed with a linear order: the order fromthe definition of a filtration determines an order on the edges joining every vertex with verticesof the previous level. The obtained graded tree will be called the filtration tree on the set A (seeFig. 1). The set of all ordered graded finite trees will be denoted by OT . Besides, we will considertrees with a marked vertex (leaf). The set of all ordered graded finite trees with a marked leaf willbe denoted by OT P . • • • | {z } • • |{z}| {z } (cid:7) |{z} • • |{z} • |{z}| {z }| {z } Figure 1: A finite filtration and its tree with a marked vertex
Let Ξ = { ξ n } n ≥ be a basic filtration on a space ( X, µ ). For n ≥ x ∈ X , consider theordered graded tree otp n ( x ) ∈ OT P corresponding to the restriction of the finite filtration { ξ i } ni =0 on the element of ξ n containing x with the marked leaf corresponding to x . By ot n ( x ) we denotethe same ordered tree without marked vertex. Consider the partition ¯ ξ n of the space ( X, µ ) intothe preimages of points under the map otp n . We say that the sequence ¯Ξ of refining partitions { ¯ ξ n } n ≥ is associated with the basic filtration Ξ. Definition 3 (see [9, 16]) . We say that a basic filtration Ξ on a space (
X, µ ) is combinatoriallydefinite if the sequence of partitions ¯ ξ n , n ≥
0, is a basis in the space (
X, µ ), that is, it converges tothe partition into singletons mod 0 (in more detail, this means that there exists a subset ˜ X ⊂ X offull measure such that for any two points x, y ∈ ˜ X there exists n such that x and y lie in differentelements of the partition ¯ ξ n ).Essentially, this definition singles out a class of basic filtrations that are completely determined,up to metric isomorphism, by the collection of invariants of their finite fragments. These invariantsare of combinatorial nature, which explains the name. The definition is inspired by another one(see [16]), which singles out a class of arbitrary (not necessarily basic) filtrations called standard .The latter definition is, in turn, a generalization of earlier work on standard dyadic and homogeneousfiltrations [9, 15]. The important question of the relation between the notions of combinatorialdefiniteness and standardness will be considered later. .2 Combinatorial equivalence of filtrations and the canonical quotient Definition 4 (see [16]) . We say that two basic filtrations Ξ = { ξ n } n ≥ and Ξ = { ˜ ξ n } n ≥ onLebesgue spaces ( X , µ ) and ( X , µ ), respectively, are combinatorially equivalent , or have thesame combinatorial type , if for every n ≥ { ξ k } nk =0 of ( X , µ ) and { ξ k } nk =0 of( X , µ ) are metrically isomorphic.It is clear from the previous definition that for combinatorially definite basic filtrations, combi-natorial equivalence coincides with metric isomorphism.Let Ξ be a basic filtration on a space ( X, µ ). Consider the equivalence relation on X determinedby the associated sequence of measurable partitions ¯Ξ: points x and y lie in the same equivalenceclass if and only if otp n ( x ) = otp n ( y ) for all n ≥
0. This equivalence relation is measurable andrespects the order, hence we can take the corresponding quotient. The resulting filtration will becalled the canonical quotient of the basic filtration Ξ.
Remark 1.
The canonical quotient of a combinatorially definite basic filtration Ξ coincides with Ξitself. The canonical quotient of an arbitrary basic filtration Ξ is combinatorially equivalent to Ξ,but, in general, not metrically isomorphic to Ξ.
Remark 2.
If Ξ is a basic filtration for an automorphism T , then the canonical quotient of Ξdetermines a quotient of T . However, unlike the quotient filtration, this quotient automorphism isnot canonical, since it depends on the choice of an approximation.The canonical projection turns the space of all basic filtrations into a bundle over the space ofall combinatorially definite basic filtrations. A very important and interesting question is whetherthe fibers of this bundle are isomorphic in some sense, i.e., whether the bundle is isomorphic toa direct product. Z and classification of combinatorially defi-nite filtrations Definition 5. A hierarchy on Z is a filtration on the space Z endowed with the counting measurethat is basic for the left shift in the sense of Definition 2. By I we denote the set of all hierarchieson Z ; this is a compact space in the natural topology.In other words, a hierarchy on Z is a coarsening sequence of partitions of Z such that everyelement of every partition is a finite interval of consecutive integers and any two integers lie in thesame element of some partition with a sufficiently large number.Let Ξ be a basic filtration on a space ( X, µ ). With almost every point x ∈ X we associate thehierarchy I ( x ) on Z determined by the restriction of Ξ to the orbit of x identified with Z in a naturalway (the point T n x , n ∈ Z , is identified with n ). Note that the hierarchy I ( x ) can be understoodas the inductive limit as n → ∞ of the ordered graded trees otp n ( x ) with marked vertices.The image of the measure µ under the map I is a shift-invariant measure on the space I . Thus,a basic filtration Ξ determines an invariant random hierarchy on Z . This measure will be called the combinatorial scheme of Ξ. A basic filtration Ξ is combinatorially definite if and only if the map I is injective mod 0. Given an invariant measure ν on the space I , there exists a space ( X, µ ) anda basic combinatorially definite filtration Ξ on this space such that ν is the combinatorial schemeof Ξ. heorem 1. In the class of combinatorially definite filtrations, the combinatorial scheme is a com-plete metric invariant.
We illustrate the introduced notion with a simplest example. A measure on the space I ofhierarchies is determined by its values on the cylinders, i.e., sets of hierarchies having a fixedstructure on a given element of the partition of level n containing 0. In the case where Ξ is a dyadicfiltration on a Lebesgue space ( X, µ ), the hierarchies I ( x ) corresponding to points x of X arealso dyadic. The only shift-invariant measure concentrated on dyadic hierarchies is uniform: allcylinders determined by elements of the partition of level n have equal probabilities. Definition 6.
We say that an automorphism T of a space ( X, µ ) is combinatorially definite withrespect to a combinatorial scheme if there is a combinatorially definite basic filtration of T withthis combinatorial scheme. Theorem 2.
Every automorphism is combinatorially definite with respect to some combinatorialscheme.
The proof of this theorem essentially follows from an analog of the lacunary theorem, seeSection 4, Corollary 1.
The collection of combinatorial schemes with respect to which an automorphism T is combina-torially definite will be called the combinatorial scheme of T . Some characteristics of this schemeare metric invariants of the automorphism. It is of interest to study the behavior of the combinatorial scheme of an automorphism withrespect to various operations (taking a derivative or integral automorphism, the product of auto-morphisms, etc.).One can easily see a similarity between this definition and that of the scale of an automorphism(see [12]), which is exactly one of the invariant characteristics of the combinatorial scheme and theautomorphism itself. In more detail this relation will be discussed elsewhere.The most interesting class consists of automorphisms with the simplest possible combinatorialscheme, the dyadic one; it is this scheme that is related to the notion of a measure-preservingautomorphism with complete scale (see [12, 16]). In a later paper [5], the notion of a standardautomorphism was defined (the term is chosen by analogy with the notion of a standard dyadicfiltration introduced in [9]); the definition involves the notion of monotone equivalence in the senseof Kakutani. Apparently, the standardness of an automorphism in the sense of [5] is close to ourcombinatorial definiteness of an automorhism with respect to the dyadic scale.The concept of the combinatorial scheme of an automorphism also covers substitutional ergodictheorems related to the scale, see [12].
As already mentioned, adic shifts on path spaces of graded graphs are important examples ofautomorphisms of measure spaces. On the other hand, in [13, 14] the first author proved that everyergodic automorphism has an adic realization.
Theorem 3 ([14]) . For every ergodic automorphism T of a Lebesgue space there is a graded graph Γ endowed with an adic structure and a central measure µ on the path space T (Γ) of Γ such that theadic shift ( T (Γ) , µ ) is isomorphic to T . he language of graded graphs is closely related to the language of basic filtrations. Let Γ bea graded graph endowed with an adic structure; the space T (Γ) of infinite paths in Γ is equippedwith the tail filtration Ξ = { ξ n } determined by the structure of the graph: two paths lie in thesame element of the partition ξ n if they coincide starting from the n th level. Let µ be a centralmeasure on T (Γ) such that almost every path has a successor and a predecessor in the sense ofthe adic order; in what follows, such a measure is said to be an essential central measure. Then Ξis a basic filtration on the space ( T (Γ) , µ ), the corresponding order being determined by the adicstructure.Let Γ be a graded graph endowed with an adic structure. With each vertex v of Γ we associatean ordered graded tree ot( v ) ∈ OT according to the following rule. With the vertex of level 0 weassociate the tree consisting of a single vertex. Then we apply the following recursive (on n , where n ≥
0) procedure. Let w be a vertex of level n + 1 in Γ, and let ( v , w ) , . . . , ( v k , w ) be all edgesleading to w from vertices of level n in the adic order (some vertices v i may be repeated). Thegraded tree ot( w ) is defined as follows: its root corresponds to the vertex w itself; there are k edgesjoining it with the vertices corresponding to v , . . . , v k (counting multiplicities), the ordered gradedtree ot( v i ), i = 1 , . . . , k , already defined at the previous step hanging from each of these verticesas a root. An order on the edges leading from the root is determined by the adic structure of thegraph Γ.The leaves of the constructed tree ot( w ) are in a one-to-one correspondence with the pathsin Γ leading to w from the vertex of level 0, and the order on them corresponds to the adic order.The graded tree ot( w ) in which the leaf corresponding to a path x to w is marked will be denotedby ot( x ).Thus we have defined a map ot from the set of vertices of the graph Γ to the set OT of orderedgraded trees; we have also defined a map, denoted by the same symbol, that sends finite pathsstarting at the vertex of level 0 in Γ to ordered graded trees with marked vertices. Definition 7.
Let Γ be a graded graph endowed with an adic structure. We say that Γ is minimal if for any two vertices v, w of the same level, the trees ot( v ) and ot( w ) are different.Let µ be an essential central measure on the space T (Γ) of infinite paths in Γ. We say that themeasure µ is minimal if almost all infinite paths differ in the graded trees corresponding to theirinitial segments (i.e., there exists a subset of full measure in T (Γ) such that for any two paths x and y from this subset there is n ≥ x [ n ]) (cid:54) = ot( y [ n ]), where x [ n ] and y [ n ] are theinitial segments of length n of x and y , respectively). Proposition 1.
A basic filtration Ξ on a space ( X, µ ) is combinatorially definite if and only if itis isomorphic to the tail filtration of a minimal graded graph endowed with an adic structure andan essential central measure on the path space.Proof. Let Ξ be a combinatorially definite basic filtration. We construct a graded graph in whichthe vertices of level n , n ≥
0, correspond to the different types of ordered trees ot n ( x ), x ∈ X .Two vertices of neighboring levels are joined by an edge if they correspond to nested ordered trees;an order on the edges entering each vertex is determined in a natural way by the order of thecorresponding tree. With each point x ∈ X we associate the infinite path in the constructed graphthat passes through the vertices corresponding to the trees ot n ( x ), n ≥
0, and edges correspondingto the embeddings of otp n ( x ) into otp n +1 ( x ). Since Ξ is combinatorially definite, the resultingmap is injective mod 0, and it sends the order of Ξ to the adic order on the graph. It follows thatthe pullback of µ under this embedding is a central measure. It is an essential central measure, ince the filtration is basic. Obviously, the tail filtration of the constructed graph endowed withthis measure and the adic order is isomorphic to the original basic filtration Ξ. The minimality ofthe graph follows from the construction: one can easily check that for every vertex v the orderedgraded tree ot( v ) is exactly the tree with which v is associated in the construction.Conversely, assume that we have a minimal graded graph endowed with an adic structure andΞ is its tail filtration. If x is a path in this graph, then ot n ( x ) = ot( x [ n ]), where x [ n ] is theinitial segment of x of length n . Since the graph is minimal, every path is uniquely determinedby the collection of graded trees otp n ( x ), n ≥
0, which means exactly that Ξ is combinatoriallydefinite.
Remark 3.
Let Γ be a graded graph and Ξ be the tail filtration of Γ. If µ is an essential minimalcentral measure on the path space T (Γ), then Ξ is a combinatorially definite basic filtration of thespace ( T (Γ) , µ ).Proposition 1 shows that only a combinatorially definite basic filtration can have an adic real-ization on a minimal graph. In the case of a basic filtration that is not combinatorially definite,the construction described in the proof of Proposition 1 determines a quotient of this filtrationisomorphic to the tail filtration of a minimal graph; essentially, this is exactly the canonical quo-tient. To construct an adic realization of a basic filtration that is not combinatorially definite, it isconvenient to use the language of colored filtrations. Definition 8.
Let ξ be a measurable partition of a space ( X, µ ). We say that ξ is a colored partition if the quotient space X/ξ is endowed with a finite measurable partition c [ ξ ], called a coloring , whichassigns colors to the elements of ξ .A basic filtration Ξ = { ξ n } n ≥ is called a colored filtration if each partition ξ n is endowed witha coloring c [ ξ n ].Let Ξ be a colored basic filtration of a space ( X, µ ). For every n ≥ x ∈ X , the ordered trees otp n ( x ) considered above also become colored: the color of a vertex oflevel i is defined as the color of the corresponding element of the partition ξ i . Let cotp n ( x ) be thecolored tree otp n ( x ) with a marked vertex. The measurable partition of the space ( X, µ ) into thepreimages of points under the map cotp n will be denoted by c ( ξ n ). Definition 9.
We say that a colored basic filtration Ξ of a space (
X, µ ) is combinatorially definite if the sequence of partitions c ( ξ n ), n ≥
0, is a basis of (
X, µ ).Let Γ be a graded graph endowed with an adic structure and Ξ be the tail filtration on thepath space T (Γ). Let µ be an essential central measure on T (Γ), not necessarily minimal. ThenΞ is a basic filtration on the space ( T (Γ) , µ ). A natural coloring of the elements of the partition ξ n , n ≥
0, is determined by the vertices of level n in Γ: assign a color to each such vertex, anddefine the color of an element of ξ n as the color of the vertex of level n lying on the paths from thiselement. The colored basic filtration thus defined will be called the canonical colored filtration ofthe graph Γ. Proposition 2.
Let Γ be a graded graph endowed with an adic structure and an essential centralmeasure. Then its canonical colored filtration is combinatorially definite. Conversely, if a Lebesguespace ( X, µ ) is endowed with a combinatorially definite colored basic filtration Ξ , then there existsa graded graph Γ and an essential central measure ˜ µ on its path space T (Γ) such that the canonicalcolored filtration of Γ is isomorphic to Ξ . roof. The proof reproduces the proof of Proposition 1.
Proposition 3.
For every basic filtration on a Lebesgue space there is a coloring such that theresulting colored basic filtration is combinatorially definite.Proof.
Let Ξ = { ξ n } n ≥ be a given basic filtration on a Lebesgue space ( X, µ ). Fix a sequence { ζ n } n ≥ of finite partitions of the space ( X, µ ) that separates points mod 0. Since Ξ is a basicfiltration, for every n ≥ ξ n is a finite ordered set of size bounded bya constant a n depending only on n . Let us regard ζ n as a coloring of the points of X into a finitenumber, say b n , colors. Then every element of ξ n is a finite ordered colored set; in total, thereare at most (cid:80) a n k =1 b kn possible colorings. Thus we have defined a coloring of the partition ξ n . Thefiltration colored in this way is combinatorially definite. Indeed, for almost any two points x, y ∈ X there is n such that x, y lie in different elements of the partition ζ n . But then the colored treesotp n ( x ) and otp n ( y ) are different.Propositions 2 and 3 essentially describe the proof of Theorem 3. The question about a real-ization of a periodic ergodic automorphism is meaningless, hence we may assume that the auto-morphism is aperiodic. In this case, we can construct a basic filtration, which is not necessarilycombinatorially definite, but, according to Proposition 3, can be colored in such a way as to be-come combinatorially definite. Proposition 2 gives a construction of an adic realization of a coloredfiltration, completing the proof of Theorem 3. In this section, we prove a strengthening of Theorem 3. Namely, we prove that all (aperiodic)automorphisms of a Lebesgue space can be realized on a single special graph endowed with an adicstructure; it suffices to vary only a central measure µ on its path space. Earlier, a similar resultwas obtained for the class of automorphisms having the dyadic odometer as a quotient: all suchautomorphisms can be realized on the so-called graph of ordered pairs (for details, see [18]). Consider the following graded graph. Level 0 contains a single vertex. Having a set V n of verticesof level n , we define a set V n +1 of vertices of level n + 1 as V n +1 = V n (cid:116) copy( V n ). Every vertex w ∈ V n is understood as an ordered pair ( u, v ) of vertices of level n , and we draw edges from u and v to w , endowing them with a natural order: the edge ( v, w ) is greater than ( u, w ). Everyvertex w ∈ copy( V n ) is understood as a copy of a vertex u of level n , and we draw a unique edgefrom u to w . The resulting graph endowed with an adic structure will be called the uniadic graphand denoted by U A (see Fig. 2).Recall the definition of a telescoping of a graded graph.
Definition 10.
Let Γ be a graded graph and { k n } n ≥ be a strictly increasing sequence of non-negative integers with k = 0. We define a telescoping of Γ as follows. The vertices of level n inthe new graph correspond to the vertices of level k n in Γ, and two vertices of neighboring levelsin the new graph are joined by an edge of multiplicity equal to the number of paths in Γ betweenthe corresponding vertices. An adic order on the edges of the new graph is determined by the adicorder on the corresponding paths in the original graph. Definition 11.
We say that a graded graph Γ is an induced subgraph of a graded graph Γ if theset of vertices and the set of edges of Γ are subsets of the set of vertices and and the set of edgesof Γ, respectively, and, besides, if v is a vertex of Γ , then Γ contains all edges of Γ coming to v from vertices of the previous level. An order on the edges is inherited in a natural way.The following properties are clear from definitions. Remark 4.
If a graded graph Γ is an induced subgraph of a graded graph Γ, then the space ofinfinite paths in Γ is a subset of the space of infinite paths in Γ invariant under the adic shift on Γ. Remark 5.
If a graded graph Γ is a telescoping of a graded graph Γ, then the adic shifts on thesegraphs are isomorphic. The uniadic graph
U A is universal in the following sense.
Proposition 4.
Let Γ be a graded graph endowed with an adic structure such that every vertex hasat least two edges entering it from above. Then there exists an induced subgraph Γ of U A suchthat some telescoping of Γ is isomorphic to Γ (with the isomorphism respecting the adic order). Proof.
To prove this, it suffices to realize that for every n ≥
0, the bipartite graph formed by the n thand ( n + 1)th level of Γ can be extended by several intermediate levels so that the resulting gradedgraph (with finitely many levels) satisfies the following two properties: every vertex of every level(except the topmost one) has either one edge or an ordered pair of edges entering it from above,and no two vertices have the same ancestors (taking into account the order); a telescoping of thisgraph coincides with the original bipartite graph.The existence of such a thinning of the bipartite graph can be proved as follows. First, byan appropriate thinning, we ensure that no level contains two vertices with the same ancestors.The rest can be easily proved by induction on the number of edges in the bipartite graph. If somevertex w of the bottom level has more than two incoming edges, say from vertices v , . . . , v k (takinginto account the order), then insert an intermediate level consisting of a copy of the top level withone additional vertex ( v , v ). Join this vertex by two upward edges with v and v , exactly in thisorder. Join all vertices of the bottom level except w with copies of vertices of the top level as in the Proposition 4 is, in a sense, akin to the lacunary theorem (see [8, 16]): an appropriately thinned filtration becomescombinatorially definite (in the lacunary theorem, standard). riginal bipartite graph. Finally, join w with the vertex ( v , v ) and the copies of the other vertices v i , i = 3 , . . . , k (see Fig. 3). The path space of the new graph is isomorphic to the path space ofthe original graph, but now it remains to construct a thinning of a bipartite graph with one edgeless. Figure 3: Thinning a bipartite graph
It is not difficult to see that in Theorem 3, given an aperiodic ergodic automorphism T , onecan require that the constructed graded graph Γ satisfies the conditions of Proposition 4 (see [14]and [19]). This proposition allows one to embed the path space T (Γ) of such a graph into T ( U A ),and thus we arrive at the following theorem.
Theorem 4 (metric universality of the uniadic graph) . For every ergodic automorphism T ofa Lebesgue space there is a central measure µ on the path space T ( U A ) of the uniadic graph U A such that the adic shift on the space ( T ( U A ) , µ ) is isomorphic to T . Corollary 1.
For every ergodic automorphism T of a Lebesgue space ( X, ν ) there is a combinato-rially definite basic filtration Ξ on ( X, ν ) .Proof. Use Theorem 4 to find a central measure µ on the path space T ( U A ) of the uniadic graph
U A such that the adic shift on the space ( T ( U A ) , µ ) is isomorphic to T . The graph U A is minimal inthe sense of Definition 7, hence, by Proposition 1, its tail filtration is combinatorially definite forthe adic shift isomorphic to T .Theorem 4 essentially states that for every aperiodic automorphism T of a Lebesgue space( X, ν ) there is a measurable mod 0 injective map f from X to the path space T ( U A ) of the uniadicgraph that sends T to the adic shift. If we fix the space X and the transformation T and vary theinvariant measure ν , then the map f described above will, in general, vary, since the constructionof an adic realization of a given automorphism depends on the measure (see [14]). But can onemake the map f independent of the measure ν ?Arguing in this way, we arrive at the following “Borel” question, which was raised in [18]. Let X be a standard Borel space and T be an (aperiodic) Borel automorphism of X . Does there exista graded graph Γ endowed with an adic structure and a Borel-measurable embedding f of X intothe path space T (Γ) that sends T to the adic shift? t turns out that the answer to this question is positive. If T is an aperiodic Borel automorphismof a separable metric space X , then one can construct such a graph Γ and such an embedding f .Moreover, by Proposition 4, the same uniadic graph U A can serve as a Borel universal graph.
Theorem 5 (Borel universality of the uniadic graph, [19]) . Let T be an aperiodic Borel automor-phism of a separable metric space X . Then there exists a Borel subset ˆ X ⊂ X such that µ ( ˆ X ) = 1 for every T -invariant Borel measure on X and a Borel-measurable injective map f from ˆ X to thepath space of the uniadic graph that sends T to the adic shift. A complete proof of this theorem will be published separately (see [19]). Here we give onlya sketch of the proof. It essentially follows the proof of Theorem 3, except that all steps of theconstruction should now be Borel, i.e., independent of the measure.The first step of the proof, as in the case of Theorem 3, is a weakening of Rokhlin’s lemma.A Borel version of Rokhlin’s lemma, unlike the classical one, seems far from being a trivial problem.For example, if we consider the shift T on the space { , } Z , it says that for every ε > B ⊂ { , } Z such that B ∩ T B = ∅ and for every T -invariant aperiodic measure µ on { , } Z , we have µ ( B ∪ T B ) > − ε . The problem of finding a measure-free proof of the lemma,i.e., proving its Borel version, was posed by V. A. Rokhlin in a conversation with the first author.A Borel version of Rokhlin’s lemma was proved by B. Weiss and E. Glasner [3, p. 628, Propo-sition 7.9]. Lemma 2.
Let T be a homeomorphism of a Polish space ( X, ρ ) . Let n ∈ N and ε > . Then thereexists a Borel subset B ⊂ X such that the sets B, T B, . . . , T n − B are pairwise disjoint and µ ( B ∪ T B ∪ · · · ∪ T n − B ) > − ε for every T -invariant aperiodic measure µ on X . To prove Theorem 5, as in the case of Theorem 3, one should first weaken this version of thelemma by dropping the periodicity condition for approximating automorphisms, and then apply itrepeatedly, passing after each iteration to the derivative automorphism on the constructed set B .In the language of filtrations, the proof consists in constructing a Borel basic filtration of theautomorphism T , coloring it, and embedding the colored filtration into the path space of theuniadic graph. For details, see [19]. References [1] S. Bezuglyi, O. Karpel,
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