The Thue-Morse shift, Baumslag-Solitar group, and biminimality
aa r X i v : . [ m a t h . G R ] O c t THE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, ANDBIMINIMALITY
LAURENT BARTHOLDI
Abstract.
Call a group action on a topological space biminimal if for anypoints x, y P X there exists a group element taking x arbitrarily close to y andwhose inverse takes y arbitrarily close to x .A symbolic encoding of the Thue-Morse dynamical system is given, in termsof ω -automata. It is used to prove that the Thue-Morse dynamical system isminimal but not biminimal.The ω -automata also establish a link between Nekrashevych’s presentationof limit spaces and solenoids with a construction described by Vershik andSolomyak. Introduction
The goals of this note are threefold: develop, on one important example, thegeneral theory of automatic actions [in preparation]; give a symbolic / topologicaldescription of the Thue-Morse shift; and answer a question of Dominik Francœurrelated to a strengthening of minimality of actions.It is my belief that salient aspects of the theory of automatic actions becomemore digestible when specialized to an example, in particular if this example isfundamental. Note that Vershik and Solomyak gave in [9] an excellent account ofthe Thue-Morse shift and its interpretation as an adic transformation; however,that description is only valid measure-theoretically; by force of some simplificationsadopted in their construction, the map they construct cannot be continuous.I elected not to present the results in the most economical or terse possible way,because part of my motivation is to establish connections with the classical notationand terminology. The fundamental messages should be: finite state automata areuseful in more than one manner, to encode in finite, easily manipulable objects vari-ous infinite sets or languages; and it helps to keep track of the group (or semigroup)acting on the space in question. To paraphrase Louis Ferdinand C´eline [1, incipit](as suggested by Jacques Sakarovich), ! Les automates, c’est l’infini mis `a port´ee des caniches " .1.1. The Thue-Morse automaton.
The Thue-Morse map, originally consideredby Thue [8], and Morse [5] in his study of geodesics on surfaces, is the endomorphism ζ of t , u ˚ is given by 0 ÞÑ , ÞÑ
10. Let Ω
Ă t , u Z denote those w P t , u Z such that all subwords of w occur in ζ n p q for some n . It is a compact, totallydisconnected set naturally carrying a Z -action by translation. Date : October 16th, 2019.
Ω may be recoded in terms of paths in a “Bratteli-Vershik diagram”, on whichthe action of Z may be defined combinatorially. However, the classical diagramis only valid measure-theoretically; there is a map Ω ։ t paths in the diagram above u which is almost-everywhere bijective, but is 2 : 1 on a countable set, on which thethe natural image of the shift map is discontinuous. The otherwise-excellent refer-ence [2] remarks dryly that “this diagram [. . . ] does not fit our setting”.On the other hand, a powerful technique to encode dynamical systems has beendeveloped by Nekrashevych [6]; in a word, “Nekrashevych duality” establishes anequivalence between certain self-similar group actions and self-coverings of spaces,which doesn’t apply to Ω – but almost does. If a picture is worth a thousand words,the main outcome of the present paper is a finite automaton: A µ : adcfb eµµµµµ µµµ µ µ c f d a b | b c | c f | f e | e d | d a | a c | a b | a c | e d | b e | f d | f b a c b e d f e b | e f | c a | d e | b The inner six states encode the Thue-Morse shift Ω: there is an easily-computablebijection between Ω and the set of right-infinite paths in this 6-vertex graph. Pathsare naturally identified with the labels read along them. The Z -action is describedby the other states of A µ : to compute the image of a path labeled w , find theunique path starting at a state labeled µ and carrying a label of the form w | w (where we interpret labels i j as i j | i j ); then the image of w is w .Call A the subautomaton of A µ spanned by the inner six vertices; denote by L p A q the space of infinite paths in A (topologized by declaring paths close if theyagree on a long initial segment). We prove: Theorem A (= Theorems 5.1 and 5.2) . There is a homeomorphism Ω – L p A q .Using it, there is a natural factor map π : Ω ։ Z , given by sending label i j to i in A µ , and mapping the shift action on Ω to the natural Z -action on -adic integers.It is generically , and on π ´ p Z q .Define r Z as a modification of Z in which the copy of Z is duplicated. Then thereis a homeomorphism Ω – Z { ¨ r Z , a -point extension of r Z , under which the shift’saction on Ω is transported to “addition with a cocycle” p s, z q ÞÑ p s ` φ p z ` q , z ` q for the map φ : r Z Ñ Z { given in (6) ; and the factor map Ω ։ Z is given by thenatural maps Z { Ñ and r Z Ñ Z . HE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, AND BIMINIMALITY 3
Denote by p L p A q the space of bi-infinite paths in A , and write w „ w for twobi-infinite paths in A if there exists a bi-infinite path in A µ labeled w | w . Weconstruct a “solenoid” for the Thue-Morse shift: a suspension of Ω on which thedynamics induced by ζ becomes invertible. Theorem B (= Theorem 6.2) . The relation „ is an equivalence relation, and thequotient space S : “ p L p A q{„ (1) is compact, metrizable, connected; (2) fibres over the circle R { Z with fibre L p A q ; monodromy around the circleinduces the shift action on the fibre; (3) admits a quotient map to the -adic solenoid S “ p Z ˆ R q{p z, t ` q „p z ` , t q , induced by forgetting alphabet decorations; fibres have cardinality or . There is a homeomorphism S ÝÑ Z { ¨ ˆ r Z ˆ r , sp z, q „ p z ` , q ˙ , on the image of which the action is given by “addition with cocycle”, andon which the map S ։ S is given by r Z Ñ Z and Z { Ñ . Binimimal actions.
The following discussion arose during Dominik Francœur’sPhD defense [3]. Let G be a group acting on a topological space Ω. Recall that theaction is minimal if every G -orbit in Ω is dense, namely if every point in Ω can betaken into any open set by an element of G .Let us call the action biminimal if this definition can be made symmetric: forevery points with open neighbourhoods x P U Ď Ω and y P V Ď Ω, there exists anelement of G taking x into V whose inverse takes y into U .Define τ : X ˆ X Ñ X ˆ X by τ p x, y q “ p y, x q . In case G is abelian, we may let G act on X ˆ X by g ¨ p x, y q “ p g p x q , g ´ p y qq , and then the the action is biminimalif τ may be approximated pointwise by the antidiagonal action of G . Question 1.1.
Is every minimal action biminimal?
An easy remark: if p G, X q is minimal and G acts on X by isometries then p G, X q is biminimal. Indeed if for all x, y P X and r ą g P G with g p x q P B p y, r q ,and then g ´ p y q P B p x, r q .Another easy remark, due to Volodymyr Nekrashevych : let p G, X q be a minimalaction and fix x, U , V as above. Define V “ Ť g P G : g p x qP V g p U q X V . Then V is opendense in V : given any W open in V there is g P G with g p x q P W so g p x q P V and W X V ‰ H . Therefore, if X is a Baire space then for every x P X the “good” y in the definition are generic (= comeagre). However, we shall see: Theorem C (= Theorem 7.2) . The Thue-Morse dynamical system is minimal butnot biminimal.
Presumably, a little more work could lead to a more general statement:
Conjecture 1.2.
Let σ be a minimal homeomorphism of the Cantor set. Theneither σ is an isometry in a metric compatible with the Cantor set’s topology, or σ is not biminimal. Indeed every minimal homeomorphism may be encoded by a Bratteli-Vershikdiagram, as we will see below; and Bratteli-Vershik homeomorphisms are classifiedinto odometers (which preserve a metric) and expansive maps (which are thus
LAURENT BARTHOLDI subshifts). These correspond to Bratteli diagrams / automata with 1, respectively ą ą The Thue-Morse shift space
Set Σ “ t , u , and recall the Thue-Morse endomorphism ζ of Σ ˚ given by0 ÞÑ , ÞÑ
10. Note that it extends to Σ N and Σ Z by continuity.Let Ω Ă Σ Z denote those w P Σ Z such that all subwords of w occur in ζ n p q forsome n . It is a compact, totally disconnected set naturally carrying a Z -action. Ingeneral, for a set F and an invariant subspace X of some F N or F Z , we denote by σ or σ X the endomorphism of that space given by σ X p u q n “ u n ` , and call thesubspace a subshift .Set u “ ζ p q “ ¨ ¨ ¨ P Σ N a right-infinite word, and for words v, w P Σ N denote by ‘ v.w ’ the word obtained by concatenating v (in reverse) with w , namely we have p v.w q n “ w n and p v.w q ´ ´ n “ v n for all n P N ; so σ ’s actionamounts to moving the ‘ . ’ one position to the right. It is easy to see that we haveΩ “ t σ n p u.u q : n P Z u . It is well-known that Ω is aperiodic , namely the Z -action is free. Moreover, thereis no subword of u of the form pqpqp , as is shown by induction on the length of p, q .In fact, a slightly larger group acts on Ω: first, there is a central involution w ÞÑ w given by the exchange 0 Ø
1. Since the infinite dihedral group D : “ x β, γ | β , γ y acts on Z by β p n q “ ´ n and γ p n q “ ´ ´ n , it also acts on Ω, and the action isfree except for two orbits, those of u.u and u.u (both stabilized by γ ); more on thislater.Recall that a dynamical system is minimal if every orbit is dense. It is well-known that our Ω is minimal, the criterion being that u.u is repetitive : for everysubword v of u.u , there is a constant C p v q such that every subword of size C p v q in u.u contains a copy of v . This property in turn follows directly from the nature of ζ , with C p v q growing linearly in | v | . Again, more on this later.Note furthermore that Ω admits another, non-invertible action given by ζ ; really,we have an action of the “Baumslag-Solitar semigroup” B p , q ` “ x α, α ´ , ζ | α ˝ ζ “ ζ ˝ α y ` , with α acting as σ and ζ acting as ζ on Ω. We could even include D in oursemigroup, and consider x β, γ, ζ | β “ , γ “ , ζα “ αζ, ζβ “ βαβζ y ` if wewanted (we don’t).We note for future use the important property Ω “ ζ p Ω q \ σζ p Ω q : indeed u doesnot contain 000 or 111 as a subword, so the same holds for every w P Ω. Thus w contains a subword 01, at either an even or odd location. Depending on thesecases, either w or σ p w q may be factored into 01 and 10 subwords.Remark that ζ ´ is a well-defined homeomorphism on ζ p Ω q . We extend it to σζ p Ω q by ζ ´ p σw q “ ζ ´ p w q . We obtain a continuous, 2 : 1 map ζ ´ on Ω,satisfying ζ ´ ˝ ζ “ ζ ˝ ζ ´ p w q P t w, σ ´ p w qu for all w P Ω. HE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, AND BIMINIMALITY 5 Bratteli diagrams
Even though the Z -action is very easy to understand on individual elements of Ω,it is not easy to study its dynamical properties. For this, it is classical to re-encodeΩ via Bratteli diagrams.We first extend the alphabet Σ into Σ : “ t , , , , u , and extendthe substitution ζ to(1) ÞÑ , ÞÑ , ÞÑ , ÞÑ , ÞÑ , ÞÑ . What we have done is “collared” the original substitution; namely if w P Ω thenwe re¨encode it as r w P Σ Z by r w p n q “ w p n ´ q w p n q w p n ` q . The corresponding mapΩ Ñ t r w : w P Ω u Ă Σ Z is clearly a homeomorphism on its image. Note that ζ naturally acts on Σ Z , so B p , q ` acts on Σ Z preserving the image of Ω.We finally encode Ω into Σ N by λ : Ω Ñ Σ N ,w ÞÑ pp ζ ´ n p r w qqp qq n P N . The image of λ is easy to understand using a graded graph V N called a Brattelidiagram , repeating periodically in an infinite stack growing upwards the followingpicture V : The diagram V is constructed as follows: every ‘ a ÞÑ bc ’ in the extension of ζ to Σis written as two edges in this diagram, going upwards from b to a and from c to a ,with an arrow from the first to the second. We will use these arrows later; for now,let us call “minimal edge” the source of the arrow, and “maximal edge” its range.Let P p V q denote the space of all upwards-going paths in V N , recorded as se-quences of vertices z “ p z , z , . . . q . The topology on P p V q declares as open sets all O p y ,...,y n q “ t z P P p V q : z i “ y i @ i ď n u , and is homeomorphic to the Cantor set.(Note that we’ll later encode paths by their edges, but here it makes no difference.) Proposition 3.1.
The map λ is a homeomorphism Ω Ñ P p V q .Proof. First, λ p Ω q Ď P p V q : indeed the diagram just says that if r w contains a letter b or c , then this letter must appear inside ζ p a q for some a P Σ, so the sequence λ p w q must follow a path in V N .Clearly λ is continuous, since the n th letter of λ p w q only depends on a finiteportion (of size 2 n around the origin) of w .Consider next a path z “ p z , z , . . . q P P p V q ; we wish to show that it hasprecisely one preimage under λ . Define ǫ i for i P N as follows: if the edge z i Ñ z i ` in z is minimal, set ǫ i “
0, otherwise ǫ i “
1. The preimage of the clopen
LAURENT BARTHOLDI O p z ,...,z n q under λ is the set of sequences w P Σ Z that contain ζ n p z n q at positions r´ ř i ă n ǫ i i , n ´ ř i ă n ǫ i i r , and in particular is non-empty; thus the intersectionof these clopens is non-empty and λ is surjective.If furthermore the sequence p ǫ , ǫ , . . . , q is not eventually constant, then theintervals above grow left and right with union Z , so the intersection of the aboveclopens is a single point and λ is injective.It remains to consider the case of p ǫ i q eventually constant, and easily reduce tothe case of constant ǫ i . For future use, call minimal , respectively maximal , a path z “ p z , z , . . . q in V N , if all its edges are so. It is easy to see that there are fourmaximal and four minimal infinite paths: dashed=minimal, solid=maximal: The four minimal paths encode sequences as follows: p , , , . . . q “ λ p ζ p q .ζ p qq , p , , , . . . q “ λ p ζ p q .ζ p qq , p , , , . . . q “ λ p ζ p q .ζ p qq , p , , , . . . q “ λ p ζ p q .ζ p qq , and the four maximal paths encode the same sequences, shifted one step left. Itis easy to see that, in this case too, the map λ is injective. Note that we neededthe “collaring” here: without it, there would be only two encodings of minimal, ormaximal, paths. (cid:3) The action of the semigroup B p , q ` on Ω can now be transported via λ to P p V q .Let us describe on P p V q the action of σ : Ω ý ; it will be an “adic” transformation µ . Clearly for every vertex in V N there is a unique minimal path ending at thatvertex, namely the one defined going downwards by always following the minimaledge.For the path p z , z , . . . q : let n be smallest such that the edge p z n ´ , z n q isnot maximal; let p z n ´ , z n q be the corresponding maximal edge. Then define µ p z , z , . . . q “ p z , . . . , z n ´ , z n , . . . q where p z , . . . , z n ´ q is the minimal path end-ing in z n ´ .In case there is no such n , this means that p z , z , . . . q is maximal, and we haveto extend µ appropriately. In fact, there is a unique continuous extension, as alimit of µ p z , . . . , z n , z n ` , . . . q for non-maximal edges p z n , z n ` q . This may be seenas follows: the first n ´ µ p z , . . . , z n , z n ` , . . . q are independent of the HE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, AND BIMINIMALITY 7 choices at positions ą n , as the following picture shows:(2) Here the path p q ω is maximal, and I drew solid paths coinciding with it onits first two or three edges; every non-maximal path has to bifurcate away from itas indicated. The corresponding successors are drawn in dashed, and they all startby a corresponding prefix of p q ω .Note therefore that the action of µ is almost finitary: except for the four maximalpaths, the paths s and µ p s q are cofinal, namely coincide starting from some pointon.Note also that it is here that we need the “collaring”: without having decoratedletters with their left and right neighbours, we couldn’t know which maximal pathgoes to which minimal one.The map ζ : Ω ý is transported via λ to a section σ ´ of the shift σ on P p V q ,and simply prepends to a path in P p V q the unique minimal edge abutting to itsstart vertex. Proposition 3.2.
The map α ÞÑ µ, ζ ÞÑ σ ´ defines an action of B p , q ` on P p V q , and turns λ into an equivariant homeomorphism.Proof. The claim is immediate, except perhaps for why σ : Ω ý is transported to µ . Now if w P ζ p Ω q then λ p w q starts with a minimal edge and µλ p w q coincideswith λ p w q except that its first edge is now maximal; thus λ ´ µλ p w q P σζ p Ω q , andis in fact σ p w q , so µ ˝ λ “ λ ˝ σ in that case. If w P σζ p Ω q then σ p w q P ζ p Ω q and ζ ´ σ p w q “ σζ ´ p w q ; and similarly λ p w q starts with a maximal edge and µλ p w q “ z z with z minimal and z “ µ p σλ p w qq , as required. (cid:3) Note that we may easily define an inverse of λ on P p V q using µ : we have λ ´ p z q “ p µ n p z q q n P Z . Automatic actions and ω -regular languages We rephrase the previous section in the more flexible language of automata.We first recall the notion of ω -regular languages . We fix once and for all a finitealphabet Σ. An ω -automaton is the data of a finite directed graph A , two subsets A ˚ , A : of its vertex set called initial and final states, and a labelling of edges of A by Σ. The ω -language that it recognizes is the following subset L p A q of theset Σ ω of right-infinite words over Σ: it consists of those w P Σ ω for which there In this section we switch from the notation Σ N to Σ ω out of deference for this standardterminology. We will switch back to N when we embed Σ N into Σ Z in the next section. LAURENT BARTHOLDI exists a path in A labeled w , starting at a vertex in A ˚ , and passing infinitelymany times through vertices in A : . A subset L Ď Σ ω that can be recognized by an ω -automaton is called an ω -regular language .For example, P p V q is recognized by the following ω -automaton in which allstates are initial and final; it is obtained by identifying the top and bottom rows in V : Note that each edge’s label is simply the label of its source vertex.
Definition 4.1.
Let L be an ω -regular language, and let a (semi)group G actingon L Ď Σ ω . The action is called regular if for every g P G its graph tp w, g p w qq : w P L u Ď L ˆ L is a regular language in Σ ω ˆ Σ ω “ p Σ ˆ Σ q ω . △ By classical properties of ω -regular languages, it is sufficient to check that thegraphs of generators are regular.We shall see that the action of µ on P p V q is regular. However, before doingso, we change once more the notation: first, we write L p A q instead of P p V q , sincewe are about to forget about the Bratteli diagram. We change our alphabet to t i j : i P t , u , j P t a, b, c, d, e, f uu the set of edges of A , as follows: we renamevertices as “ a, “ b, “ c, “ d, “ e, “ f so as to avoid multiple subscripts, and label the minimal edge ending at vertex j as 0 j and the maximal one as 1 j . We thus get A : a dc fb e c f d a b a c b e d f e Lemma 4.2.
The ω -language accepted by A is P p V q .Proof. The translation is direct: the states of A are in bijection with the verticesof V , and every edge ‘ v Ñ w ’ in V gives a transition in A from v to w , labeled‘0 w ’ or ‘1 w ’ according to whether the edge is minimal or maximal. (cid:3) Let us now construct an automaton recognizing the action of µ . We shall in factadd to A new states µ jk for all states j, k of A , representing the action of µ onpaths starting at j when their image starts at k . The labels of the edges are written‘ e | f ’ rather than p e, f q for e, f in our alphabet t i j u ; and even ‘ e ’ rather than p e, e q .Our new automaton will in fact contain the previous one; choosing as initialstates all j P t a, b, c, d, e, f u yields L p A q , and in other words the action of the HE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, AND BIMINIMALITY 9 identity if we identify L p A q with its diagonal in L p A q ˆ L p A q . Choosing as initialstates all states labeled µ jk yields the action of µ . All states in A µ are final. Werecall: Proposition 4.3 (e.g. [7, Proposition 3.7]) . An ω -regular language L is closed in Σ ω if and only L “ L p A q for an automaton A in which all states are final. Thus we recover that L p A q is compact, and the transformation defined by A µ is continuous.The information in the Bratteli diagram can be translated as follows: if there areminimal and maximal edges j Ñ ℓ and k Ñ ℓ respectively in the Bratteli diagram,then the automaton has a transition from µ jk to ℓ labeled 0 ℓ Ñ ℓ . It also has sometransitions from µ km to µ ℓn , but these are not entirely prescribed by the Brattelidiagram (remember the required argument about continuity!). In fact, to determinesuch edges, we must consider in A all possible continuations of the path startingwith the edge 1 k , once it reaches an edge of the form 0 p replace it by 1 p , and followbackwards the 0 q -edges. Thus for example there is a path 1 a b . . . a b c , whichgoes under µ to 0 d e . . . d a c , so there is an edge from µ be to µ ad labeled 1 a | d .Writing labels ‘ i j ’ for ‘ i j | i j ’ to highlight the previous automaton as a subautomaton,we get A µ : adcfb eµ ca µ ba µ ce µ ad µ be µ df µ ef µ db µ fc µ eb c f d a b | b c | c f | f e | e d | d a | a c | a b | a c | e d | b e | f d | f b a c b e d f e b | e f | c a | d e | b The initial states are all those labeled ‘ µ jk ’, and all states are final. Note that thetransducer describing µ is “bounded” in the following sense: for every n P N , thereis a bounded number (at most 12) paths of length n that do not reach an identitystate ( a, . . . , f ).!!Also for economy, we omit from the automaton all states that are not accessible (cannot be reached from an initial state) or not co-accessible (cannot be followed bya path traversing final states infinitely often). Automata can be minimized by fur-thermore identifying indistinguishable states. The minimal automaton associatedwith an ω -regular language is unique.Of course the automaton describing µ ´ is obtained by changing the labels ‘ e | f ’to ‘ f | e ’. Automata for µ n , with arbitrary n P Z , may be obtained by composingthe transducers in the usual way. Lemma 4.4.
The automaton A µ defines the homeomorphism µ on L p A q .Proof. First, let us check that the relation defined by A µ is a homeomorphism. Forthis, just keep the input labels on each edge of A µ , and note that the resulting automaton unambiguously minimizes to A ; and similarly when only keeping theoutput labels.Next, note that the automaton really does the following: while a maximal edgeis read, print a minimal one and repeat. The minimal edge to be printed followsfrom the computation in diagram (2). (cid:3) Note that transducers may also describe non-invertible transformations, andeven relations. For example, the map ζ , which prepends to a path its minimaledge, is defined by the following automaton A ζ (changing the initial states leadsto σζ , the map prepending to each path its maximal edge; and switching inputand output leads to the shift map σ on paths); so the action of the semigroup B p , q ` on L p A q is automatic. The automaton has stateset the alphabet, andon each transition reads a letter, printing the previously-stored one; so if A has atransition from i to j labeled ‘ z j ’ then A ζ has for all y P t , u a transition from y i to z j labeled ‘ z j | y i ’:(3) A ζ : ζ a ζ d ζ c ζ f ζ b ζ e f | c a | d f | c a | d a | b c | b b | c e | d d | e f | e b | a e | f c | a d | f c | a d | f b | a a | b c | b b | c e | d d | e f | e e | f It is instructive to compare the above automaton with the adic transformationgiven in [9, Equation (2)]: in our language, it is given by the transducer(4)
E MM MM | | | | | | | | It does not define a continuous self-map of t , u ω ; according to taste, M may beconsidered to be a discontinuous self-map (with discontinuity locus tp q ω , p q ω u );or a relation that it two-valued at these points and defines a map elsewhere; ora map that is well-defined an continuous on L : “ t , u ω ztp q ω , p q ω u , if onedeclares only the identity state E to be final. See below for the connection between µ and M . HE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, AND BIMINIMALITY 11 The covering map to -adics The substitution ζ has fixed length (all images of letters have length 2), ormore pedantically said the substitution ζ factors via t , u ÞÑ t‚u to the 1-lettersubstitution ‚ ÞÑ ‚‚ . Its Bratteli-Vershik diagram has one vertex and two edges;the corresponding automaton is B : | | τ of t , u ω given by Bratteli-Vershik dynamics is the odometer;adding it to the automaton above, we get(5) B τ : τ | | | | ω -language L p B q is identified with Z , and under this identification τ p z q “ z `
1. Define also on Z the doubling map ζ p z q “ z , and note that p α ÞÑ τ, ζ ÞÑ ζ q gives an action of the semigroup B p , q ` on Z , as a group of affine maps. Theautomaton giving ζ first prints a ‘0’, memorizing the just-read letter, and thencopies its input delayed one time unit (compare with (3)): B ζ : ζ | | | | Theorem 5.1.
The Thue-Morse system factors onto the odometer: there is acontinuous map π : L p A q ։ Z that interlaces the actions of B p , q ` .Proof. This is immediately checked on the automata: if one forgets letter decora-tions by replacing letter ‘ i j ’ by ‘ i ’, there exists a “morphism of automata”, namelyan initial-state-, final-state- and label-preserving graph morphism, from A to B and from A µ to B τ ; so τ ˝ π “ π ˝ µ . The same statement may be checked for ζ ,either via automata or directly. (cid:3) Much can be said about that factor map π . First, fibres typically have cardinality2; though 0 ω and 1 ω have 4 preimages. Note that these are precisely the elementsof Ω with an additional symmetry, i.e. on which D does not act freely.Let us be more precise. First, there is an order-2 symmetry in L p A q , given onits automaton by a half-turn. One might therefore consider rather an intermediatequotient between L p A q and Z , of the form Z { ¨ Z ; the map L p A q Ñ Z { ¨ Z is given by z ÞÑ p s p z q , π p z qq with s p z q “ z starts in t a, d, e u and s p z q “ z starts in t b, c, f u . (Going back to the interpretation of states as letters x y z ,we use for s p z q the letter y of the initial vertex of z ). There remains the issue ofeventually-minimal and -maximal paths; those are the paths mapping under π to Z Ă Z . We thus replace Z by r Z : “ p Z z Z q \ p Z ˆ Z { q , a topological space with a free action of Z in which the topology is defined bydeclaring as open neighbourhoods of p , t q P Z ˆ Z { Ů n ě N n ` t p Z ` q \tp , t qu for N P N , and of course their Z -translates. In other words, numbers withbinary representation n P Z are close to p n , n mod 2 q P Z ˆ Z {
2. There isan obvious map r Z Ñ Z given by the identity on Z z Z and p n, t q ÞÑ n on Z ˆ Z { Z -action is given on r Z by r τ p z q “ z ` r τ p z, t q “ p z ` , t q . The non-invertible dynamics τ : Z ý also lift to r Z : one defines a map r ζ on r Z by r ζ p z q “ z and r ζ p z, t q “ p z, t ` q . We thus have an action of B p , q ` on r Z , compatiblewith the map r Z Ñ Z .We can then improve the map π into a map r π : L p A q Ñ r Z as follows: if z P L p A q is path not ending in 0 ω or 1 ω , then r π p z q is the element π p z q of Z z Z read along the labels of the path. If however z ends in 0 ω or 1 ω , then r π p z q is p π p z q , t q with t “ t b, e u at arbitrarily large even positions, and t “ t b, e u at arbitrarily large odd positions. Theorem 5.2.
The map z ÞÑ p r π p z q , s p z qq is a homeomorphism between L p A q and r Z ˆ Z { , and the homeomorphism µ translates, via this homeomorphism, to“addition with a cocycle” r µ : p z, s q ÞÑ p z ` , s ` φ p z ` qq ; The cocycle φ : r Z Ñ Z { is given by (6) φ p n p Z ` qq “ φ p n p Z ` q , t q “ n ` , φ p , t q “ t ` . In terms of binary expansions, the cocycle is given by φ p n q “ φ p n , t q “ n ` φ p ω , t q “ t `
1. Let us state a fundamental property of A ,which comes from an analogous statement for the substitution ζ and which wehave already implicitly used: Lemma-Definition 5.3.
Given a sequence w P t , u ´ N and a state h P A , thereis a unique reverse path p w,h : ´ N Ñ A ending in h and whose labels project to w under forgetting decorations. (cid:3) Proof of Theorem 5.2.
First, the map p r π, s q is continuous: on paths not ending in0 ω or 1 ω , the first n bits of its output depend only on the first n edges. For pathsending in 0 ω or 1 ω , the paths crossing t b, e u at even, respectively odd positions arein disjoint clopens.We turn to bijectivity of r µ . Given a “symmetry” bit s and a bit sequence x “ x x . . . representing an element of r Z (with an additional bit in case thesequence is ultimately constant): consider for large m P N the six reverse paths p ´ ω x ...x m ,h given by Lemma-Definition 5.3. If x starts by n ` then these pathsmust start in t c, d u and the symmetry bit decides which; if x starts in n a similar(slightly more complicated) reasoning holds; if x P t ω , ω u then the additional bitdetermines where the preimage path starts. In all cases, the starting state of thepreimage of p x, s q is uniquely determined, and its successive states are determinedby induction.We next show that µ is carried to r µ . Without loss of generality: consider a path p starting at some vertex in t a, d, e u , i.e. with s p z q “
0. If p starts with labels e n ˚ ˚ then µ maps it to a path starting at b ; if p starts with a n ` ˚ ˚ then µ maps it to a path starting at d ; and if p starts with label d ˚ then µ maps it to a HE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, AND BIMINIMALITY 13 path starting at b or f ; so if p starts with 1 n s must beflipped while if p starts with 1 n ` s should not be changed.Consider finally the action on p x, t q P Z ˆ Z {
2, again with “symmetry” bit s “ x ‰ ´
1, then the same argument as above holds. If x “ ´
1, then the path above pp x, t q , q is either p a b q ω or p e f q ω . In the first case, t “ µ is p d e q ω , in the second case, t “ µ is p b c q ω . (cid:3) The connection between µ and the map M from (4) is via the “difference oper-ator” D : t , u ω Ñ t , u ω , given by D p x , x , . . . q “ p x ` x mod 2 , x ` x mod 2 , . . . q ;it is a 2 : 1 map implemented by the automaton D D | | | | Z with t , u ω , the identity τ ˝ D “ D ˝ M .6. Natural extensions and limit spaces
The dynamical system p L p A q , σ q “ p P p V q , σ q admits a natural extension : atopological space X equipped with a map X ։ L p A q and a self-homeomorphisminducing σ on L p A q , universal for these properties. It may be constructed as p L p A q : “ proj lim p L p A q , σ q , namely the space of sequences p z , z ´ , . . . q P L p A q ´ N such that σ L p A q p z i q “ z i ` for all i ď ´
1. The one-sided shift on p L p A q is bijective, and there is a natural map p L p A q Ñ L p A q given by p z , z ´ , . . . q ÞÑ z which interlaces the shifts σ p L p A q and σ L p A q .Recall that L p A q is a subset of Σ N for Σ “ t a , a , . . . , f , f u , and σ is as usualinduced by the shift on Σ N ; in fact L p A q is the set of ω -paths in the graph A .Thus p L p A q is naturally the set of two-sided infinite paths in A , a subset of Σ Z .Consider the “Baumslag-Solitar group” B p , q “ x α, ζ | α ˝ ζ “ ζ ˝ α y . We have an action of B p , q on p L p A q , defined as follows: ζ is the inverse of theshift σ on p L p A q Ă Σ Z . Using the paths p w,h from Lemma-Definition 5.3, thehomeomorphism µ of L p A q induces a homeomorphism p µ of p L p A q by p µ p p w,h .q q “ p w,h .µ p q q where h is the initial vertex of µ p q q , and we let α act as p µ . Lemma 6.1.
The above defines an action of B p , q on p L p A q . Proof.
It suffices to check the relation. Since we will need it later, here is theautomaton computing µ (only half of it is drawn, the other half is symmetric):(7) A µ : adcfb eµ ca µ ba µ ce µ ad µ be µ df µ ef µ db µ fc µ eb µ ca µ ba µ ce µ ad µ be c f d a b | b c | c f | f e | e d | d a | a c | a b | a c | e d | b e | f d | f b a c b e d f e b | e a | d ˚ c |˚ a ˚ b |˚ a ˚ b |˚ e ˚ a |˚ d ˚ c |˚ e Consider x “ p w,h .q P p L p A q , with q “ q r for an edge q starting at h . Then σ p µ p x q “ p w,h q .µ p r q for an edge q starting at h which is minimal if and onlyif q is minimal; writing s P t , u be the label of q without its decoration, and h for the initial vertex of µ p r q , we get σ p µ p x q “ p ws,h .µ p r q . On the other hand, p µσ p x q “ p µ p p w,h q .r q “ p µ p p ws,h .r q with h the initial vertex of r , since q is minimalif and only if q is minimal, so p µσ p x q “ p ws,h .µ p r q . We thus have σ ˝ p µ “ p µ ˝ σ . (cid:3) Solenoids.
We embark on a quick detour of the 2-adic solenoid. We start bythe short exact sequence0
Z Z r s Z r s{ Z , and apply Pontryagin duality (the dual of an Abelian group A is p A : “ Hom p A, R { Z q )to obtain 0 Z S R { Z . Here S , the Pontryagin dual of the discrete group Z r s , may be defined as p Z ˆ R q{ Z , with antidiagonal action of Z on Z ˆ R , given by n ¨ p z, x q “ p z ` n, x ´ n q .It is thus the suspension (a.k.a. mapping torus) of Z , namely p Z ˆ r , sq{p z, q „p z ` , q . Recall that a group is discrete if and only if its dual is compact, andthen is torsion-free if and only if its dual is connected; and a group is separable ifand only if its dual is metrizable.The Baumslag-Solitar group acts diagonally on Z ˆ R , by the usual affine action: α p x q “ x ` ζ p x q “ x , with x P Z or x P R . The beauty is that ζ is acontraction on Z while an expansion on R , so it induces on Z ˆ R , and hence on S , a hyperbolic map. On the suspension p Z ˆ r , sq{p z, q „ p z ` , q we see α p z, x q “ p z ` , x q , ζ p z, x q “ p z, x q if x ď { , p z ` , x ´ q if x ě { . Note that S is not the natural extension of p Z , ζ ´ q (which we identified with t , u Z ), but is a quotient of it. Quite to the contrary of natural extensions, thereis a map in the opposite direction, Z Ñ S , given by z ÞÑ p z, q . We prefer to HE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, AND BIMINIMALITY 15 consider a conjugate embedding z ÞÑ p z, q ; it interlaces the action of B p , q ` on Z with that of B p , q on S , via the obvious inclusion B p , q ` Ă B p , q .In summary, we have a space S which fibres over the circle R { Z with fibre Z and is foliated by real lines; parallel transport along the circle induces on the fibre Z the dynamics τ ; and the contracting dynamics ζ on Z may be combined withthe degree-2 covering on the circle to yield a hyperbolic homeomorphism.More precisely, by “combined” we mean that locally S decomposes as a productof a stable (Cantor set) direction and an unstable (real interval) direction; thisdecomposition is preserved by the dynamics ζ . Consider the fixed point p , q . On astable direction p Z , q of ζ which corresponds to the fibres of the covering, the map ζ ´ is a well-defined expanding map, while on the unstable direction 0 ˆp´ { , { q which is a local section of the covering, the map ζ expands by a factor of 2.6.2. The Thue-Morse solenoid.
We are about to construct a space S admittingmuch of the properties of S mentioned in the previous paragraph: it will be atopological space fibering over the circle and foliated by real lines, equipped withan action of B p , q , containing a copy of p L p A q , µ q as a fibre, such that the mon-odromy around the circle induces the map µ on the fibre, and will be a quotient of p L p A q . Furthermore all these maps are equivariant with respect to the actions of B p , q , respectively B p , q ` .In fact, for the construction of S a sizeable part of Nekrashevych’s theory of“iterated monodromy groups” may be used. The point is that the automaton N : “ A Y A µ Y A µ ´ is “nuclear”: for every n P Z , the recurrent states of theautomaton A µ n are contained in N . By induction, it suffices to check this propertyfor n “
2, and this is given by the automaton (7).We may thus define the space S : it is the quotient of p L p A q by “asymptoticequivalence”, the relation in which one declares two bi-infinite paths w, w to beequivalent if there exists a bi-infinite path in N labeled w | w . Theorem 6.2.
The space S “ p L p A q{„ (1) is compact, metrizable, connected; (2) fibres over the circle R { Z with fibre L p A q ; monodromy around the circleinduces the map µ on the fibre; (3) admits an action of B p , q in which ζ is “hyperbolic”: locally S decomposesas a product V s ˆ V u in “stable” and “unstable” directions compatible withthe action of B p , q ; the action of ζ is contracting on the V s (which arecontained in fibres of the fibration) and expanding on the V u (which aresections of the fibration); (4) admits a quotient map to S , induced by forgetting alphabet decorations;fibres have cardinality or . There is a homeomorphism S ÝÑ Z { ¨ ˆ r Z ˆ r , sp z, q „ p z ` , q ˙ , on the image of which the B p , q -action is given by “affine action withcocycle”, and on which the map S ։ S is given by r Z Ñ Z and Z { Ñ .Furthermore, all the maps are equivariant with respect to the available actions of B p , q or B ` p , q .Proof. First, let us check that the relation “ w „ w ô there exists a bi-infinite pathin N labeled w | w ” is an equivalence relation: this follows because N is nuclear. Indeed only transitivity needs to be checked; assume there is in N a path labeled w | w and a path labeled w | w . There is then a path p labeled w | w in N . Thispath can never enter a state labeled µ or µ ´ , since in N there is no edge enteringsuch states; so p remains in states µ, , µ ´ and thus lies in N .(1) The set of paths in N defines a closed subset of p L p A qˆ p L p A q , since all statesof N are final (see Proposition 4.3), so the quotient is compact and metrizable. Itsconnectedness will follow from (2), since the circle is connected and the action onthe fibre is minimal.(2) There is a natural map S Ñ J given by restriction and forgetting-of-decorations to the labels on the negative part: p w,h .q ÞÑ w . This defines amap to a quotient J of t , u ´ N . Now whenever w “ ´ ω x ´ n ¨ ¨ ¨ x ´ and w “ ´ ω x ´ n ¨ ¨ ¨ x ´ there is a path in A µ labeled w | w and ending in a state h Pt a, . . . , f u ; and conversely every path ending in such a state h has a label of thisform. There are finally paths labeled 1 ´ ω | ´ ω ending in µ ij . These are preciselythe identifications between binary sequences representing the same element in R { Z ;so J “ R { Z and the fibering map is p w,h .q ÞÑ ř n ă w n n .Every fibre is naturally identified with a right-infinite path in A , namely withan element of L p A q . The monodromy action along the circle is given by the aboveleft-infinite paths ending in a state µ ij , since they correspond to the identification1 “ R { Z ; these paths continue to right-infinite paths giving the action of µ on L p A q .(3) We first check that the action of B p , q on p L p A q descends to S . This isobvious for ζ , since „ is shift-invariant. For p µ , consider sequences p w,h .q „ p w ,h .q ,and write p µ p p w,h .q q “ p w,k .µ p q q . If q “ q , then there is a left-infinite path in N ending at h “ h and labeled w | w ; then p µ p p w ,h .q q “ p w ,k .µ p q q „ p w,k .µ p q q ; whileif q ‰ q then q “ µ ˘ p q q and again there is a path in N ending at a state µ ˘ and labeled w | w ; and then p µ p p w ,h .q q “ p w ,k .µ p q q “ p w ,k .µ ˘ p q q „ p w,k .µ p q q .Now the dynamics induced by ζ on left-infinite paths is p ...w ´ w ´ ,h ÞÑ p ...w ´ ,h ,and induces angle doubling on R { Z , so is expanding; while on the invariant fibre L p A q it is z ÞÑ h z , prepending an edge labeled ‘0’ to paths, so is contracting.The space S is covered by 6 open sets, namely for all h P t a, . . . , f u the set ofall paths passing through h at time 0, and decomposes on each such subset asa product of stable and unstable varieties: through p w,h .q they are respectively V s “ t p w,h .q : q P L p A q starting at h u and V u “ t p w ,h .q : w P t , u ´ N u .(4) The solenoid S may be viewed as the quotient of t , u Z by the “asymptoticequivalence” relation given by (5). The alphabet map i j ÞÑ i and the “morphismof automata” A µ ։ B τ induce the map S Ñ S .Now S and S are both suspensions, respectively of Z and Z { ¨ r Z , so the lastclaim follows. (cid:3) We add the remark that, for free Abelian groups, Pontryagin and Nekrashevychduality are essentially the same (or more precisely dual). Let us highlight thetautologies involved: let A be an Abelian group, equipped with an expanding self-map T : A ý . On one hand, one constructs an automatic action of A on p A { T p A qq N by acting on coset spaces A { T n p A q (which correspond to clopens in p A { T p A qq N ).Expansivity of T implies the existence of a finite automaton N (the “nucleus”)containing the recurrent subautomaton of the action of every element of A . HE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, AND BIMINIMALITY 17
Bi-infinite paths in N define an equivalence relation on p A { T p A qq Z , the quotientof which is a solenoid S “ { A r T ´ s . A leaf in S is a real vector space V , on which A acts by translation. Left-infinite paths in N define an equivalence relation on p A { T p A qq ´ N , the quotient of which is a torus, homeomorphic to V { A . We naturallyhave A “ π p V { A q , and V is the universal cover of V { A .On the other hand, Pontryagin duality associates with A the torus V ˚ { A K , sincea representation of A in R { Z extends uniquely to V , and the orthogonal lattice of A in V ˚ is by definition the set of representations that are trivial on A .Nekrasheych’s theory also applies also to non-Abelian groups G ; given a group G , a finite-index subgroup H and a homomorphism θ : H Ñ G , one obtains (bychoosing a transversal of H in G ) an action of G on p G { H q N . If θ is contracting,then this action is automatic, and also admits a finite nucleus. The “limit space”is p G { H q ´ N {„ , a compact, metrizable space X equipped with a self-covering map σ induced by the shift. One shouldn’t expect G to act on X , but the fundamentalgroup of X acts on iterated fibres of σ by monodromy, and lets one recover in thismanner the action of G on p G { H q N .In our situation, we cannot directly apply this construction: our action of Z on L p A q does not permute clopens as is the case above; and the “asymptoticequivalence relation” defines different kinds of “glue”, depending on the state of N in which the left-infinite path in N ends; different kinds of glues should be appliedto different portions of the space of left-infinite paths in A . Said differently, the bestone can hope for is a space X equipped with a collection of partial self-coverings.In our case, the space X is b e adfc and there is a self-map ζ : X ý , which is almost a self-covering, but has branchingpoints at all four vertices of X . It is given on the edges by a ÞÑ db etc. as in (1): e b ad cf The reason it is not really a covering is that the connections t b, c u Ñ t a, e u in X should not always be present; indeed A µ identifies p ω ,b and p ω ,a only whenfollowed by 0 c | c , identifies p ω ,b and p ω ,e when followed by 1 a | d , identifies p ω ,c and p ω ,a when followed by 0 b | b , and identifies p ω ,c and p ω ,e when followed by0 f | f .6.3. K-theory.
It is often enriching, for example so as to classify them, to computethe K-theory (a.k.a. dimension groups) of dynamical systems. These may be definedfrom the C*-algebra generated by the dynamical system, but we avoid all details to jump to our case of a self-map σ of a totally-disconnected compact space Ω; its K group may be defined as K p Ω , σ q “ t f : Ω Ñ Z continuous u f „ f ˝ σ , equipped with a “positive cone” K ` , the image of t f : Ω Ñ N u , and a distinguishedelement f ”
1; see [4, Theorem 1.4]. We compute it, using [4, Theorem 5.4], as K p L p A q , σ q “ inj lim p Z , A q “ Z ˆ Z r s ;here the Z corresponds to the 6 vertices of A , and in the colimit the ‘ A ’ is theadjacency matrix of A . The positive cone is K ` “ t u Y p Z ˆ Z ` r sq .By classical results, K p Ω , σ q is also the Z -equivariant ˇCech cohomology of thesolenoid p S , p µ q . 7. Biminimal actions
It is easy to see on A that L p A q is minimal: we are to check that the operation“take the successor” is transitive on finite paths. This is just the connectedness of A , namely that there does not exist any proper subautomaton. Definition 7.1.
Let G be a group acting on a topological space Ω. The action is biminimal if @p x, y P X q@p U , V open in X qDp g P G q : p x P U , y P V q ñ p g p x q P V , g ´ p y q P U q . Rephrasing, for every x, y P X and respective open neighbourhoods U , V , thereexists g P G taking x into V and U over y . △ Here and below, for s P t a, . . . , f u we use the notation ‘ sL p A q ’ to denote thoseelements of L p A q that start by the vertex s . Theorem 7.2.
The action of Z “ x µ y on L p A q is not biminimal.More precisely, consider x “ p e d q ω and y “ p d e q ω two minimal paths; andsurround them by the open sets U “ eL p A q and V “ dL p A q . Then we claim thatthere is no n P Z with µ n p x q P V and µ ´ n p y q P U . The proof will occupy the remainder of this section. For z P L p A q and s Pt a, . . . , f u , define Λ p z, s q “ t n P Z : µ n p z q P sL p A qu ;then t Λ p z, a q , . . . , Λ p z, f qu forms for all z P L p A q a partition of Z , and our claimamounts to checking that ´ Λ p x, d q X Λ p y, e q “ H .Let δ : Σ Ñ t , u be the map i j ÞÑ i that forgets letters’ decorations. Write z “ z z ; then we have equationsΛ p z, s q “ ğ p s Ñ t qP Σ p z , t q ` δ p s Ñ t q ´ δ p z q . Indeed if z is minimal then µ n p z q “ w µ n p z q with w minimal and µ n ` p z q “ w µ n p z q with w maximal; and similar equations hold if z is maximal. HE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, AND BIMINIMALITY 19
We thus have equationsΛ p x, a q “ p p y, b q ` q Y p p y, c q ` q , Λ p x, b q “ p p y, c qq Y p p y, a q ` q , Λ p x, c q “ p p y, b qq Y p p y, f qq , Λ p x, d q “ p p y, a qq Y p p y, e qq , Λ p x, e q “ p p y, d qq Y p p y, f q ` q , Λ p x, f q “ p p y, d q ` q Y p p y, e q ` q , and exactly the same equations with x, y interchanged. These determine the setsΛ p x, s q and Λ p y, s q once one specifies the “initial values”0 P Λ p x, e q , ´ P Λ p x, b q , P Λ p y, d q , ´ P Λ p y, a q . Let us change notations, and read integers in binary, LSB first (positive numbersend in 0 and negative ones end in 1 ); then the following automaton recognizesΛ p x, s q and Λ p y, s q when given initial state s , and for appropriate choices of finalstates depending on whether we wish to recognize Λ p x, s q or Λ p y, s q : c fde ade bce bac bcf bfa d
00 01 111011 0001 0011 1001 1101 001100 110000 11 10 11001001 00101111 10 0001 (The shaded state de recognizes Λ p x, d q Y Λ p x, e q , etc.) Since we are interested inrecognizing integers, we only accept infinite strings over t , u that end in ω Y ω .The remaining data that need be specified are that, to accept Λ p x, s q , one shouldstart at state s in the above diagram and accept only ω at state ade and ω atstate bcf ; while to accept Λ p y, s q one should accept only ω and ω at ade .Thus for example in terms of regular expressions Λ p x, d q “ p q ˚ E ω Yp q ˚ ˚ E ω , where E “ ε Y ˚ ˚ E is the set of words with an even numberof ’s (“evil numbers”); here is the corresponding automaton, with initial and finalvertices indicated by free incoming (respectively outgoing) arrow, and ǫ denotes an empty transition: Λ p x, d q
00 01 0011 1001 ǫ ǫ We also easily compute Λ p y, e q :Λ p y, e q
11 0001 0011 1001 ǫǫ Now it is easy to compute, using the automata, ´ Λ p x, d q ; it just amounts toswitching ω and ω : ´ Λ p x, d q
00 01 0011 1001 ǫ ǫ From these descriptions, it is easy to see p´ Λ p x, d qq X Λ p y, e q “ H , since stringsin ´ Λ p x, d q start by n ` while strings in Λ p y, e q start by n . Thus there areno n P Z with µ n p x q P dL p A q and µ ´ n p y q P eL p A q , and Theorem 7.2 is proven. HE THUE-MORSE SHIFT, BAUMSLAG-SOLITAR GROUP, AND BIMINIMALITY 21
We may translate these calculations back to the original presentation of L p A q as a subshift Ω Ă t , u Z : the sequences x, y are x “ u.u “ ζ p y q , y “ u.u “ ζ p x q , surrounded by respective neighbourhoods U “ t˚ . ˚u and V “ t˚ . ˚u .8. Acknowledgments
I am grateful to Dominic Francœur and Volodymyr Nekrashevych for discussionson these topics.
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