Extensions of Full Shifts with Group Actions
aa r X i v : . [ m a t h . D S ] J a n EXTENSIONS OF FULL SHIFTS WITH GROUPACTIONS
BARTOSZ FREJ, DAWID HUCZEK
Abstract.
We give a sufficient condition for a symbolic topolog-ical dynamical system with action of a countable amenable groupto be an extension of the full shift, a problem analogous to thosestudied by Ashley, Marcus, Johnson and others for actions of Z and Z d . Introduction
A well-known result in the study of symbolic dynamical systemsstates that any subshift of finite type (SFT) with the action of Z andentropy greater or equal than log n factors onto the full shift over n symbols – this was proven in [7] and [1] for the cases of equal andunequal entropy respectively. Extending these results for actions ofother groups has been difficult, and it is known that a factor maponto a full shift of equal entropy may not exist in this case (see [2]).Johnson and Madden showed in [5] that any SFT with the action of Z d , which has entropy greater than log n and satisfies an additionalmixing condition (known as corner gluing), has an extension which isfinite-to-one (hence of equal entropy) and maps onto the full shift over n symbols. This result was later improved by Desai in [3] to showthat such a system factors directly onto the full shift, without theintermediate extension.In this paper we use similar methods to show that in the case ofamenable group actions, any symbolic dynamical system with entropygreater than log n which satisfies a mixing condition (the gluing prop-erty , see definition 5.1), has an equal-entropy symbolic extension whichfactors onto the full shift over n symbols.2. Amenable groups and invariance
Definition 2.1.
A countable group G is called amenable if there existsa sequence ( F n ) of finite subsets of G (known as Følner sets) such that Date : January 7, 2019.2010
Mathematics Subject Classification.
Primary 37B10; Secondary 37B40.
Key words and phrases. countable amenable group, (dynamical) tiling, free ac-tion, topological entropy. for every g ∈ G we have lim n →∞ | gF n △ F n || F n | = 0 , where |·| denotes the cardinality of a set, and △ denotes the symmetricdifference. Definition 2.2.
For a pair of finite sets
T, D ⊂ G and δ >
0, we saythat T is ( D, δ )-invariant, if | DT △ T || T | < δ .Note that if D contains the neutral element of G , then the abovecondition simplifies to | DT \ T || T | < δ . Definition 2.3. If T and D are two finite subsets of G , the D -core of T is the set T D = { t ∈ T : Dt ⊂ T } . It is easy to check that for every ε > δ > T is ( D, δ ) invariant, then | T \ T D | < ε | T | , i.e., T D is a relativelylarge subset of T .3. Symbolic dynamical systems and entropy
Let Λ be any finite set. The full G -shift over Λ (often referred toas just the full shift over Λ) is the dynamical system with the spaceΛ G endowed with the product topology, and the action of G definedas ( gx )( h ) = x ( hg ). A symbolic dynamical system over Λ is any closedsubset of Λ G invariant under the action of G . If T ⊂ G is a finite set,then a block with domain T is a mapping B : T → Λ. If x ∈ Λ G and T ⊂ G is a finite set, then by x ( T ) we understand a block B withdomain T such that for every t ∈ T , B ( t ) = x ( t ). In a slight abuse ofnotation, we will not distinguish between two blocks if their domainsdiffer only by translation, i.e. for any g we treat x ( T ) and x ( T g ) asthe same block.If X is a symbolic dynamical system over Λ, then we say that a block B with domain T occurs in X , if x ( T ) = B for some x ∈ X . Finally, ifwe denote by N T ( X ) the number of blocks with domain T which occurin X , we can calculate the topological entropy of X as the limit h ( X ) = lim n →∞ | F n | log |N F n ( X ) | (where log means logarithm with base 2). It is known that this limitalways exists and does not depend on the choice of the Følner sequence(see Theorem 6.1 in [6]). In fact, we will make use of the followingconsequence of the cited theorem: Proposition 3.1.
For any ε there exists an N and δ such that if T isan ( F n , δ ) -invariant set for some n > N , then N T ( X ) > ( h ( X ) − ε ) | T | . XTENSIONS OF FULL SHIFTS WITH GROUP ACTIONS 3
We will also assume, that for every finite set D we have D ⊂ F n forsufficiently large n . 4. Tilings
We briefly recall the notions and most important results concerningtilings of amenable groups; for details we refer the reader to [4].
Definition 4.1. A tiling of an amenable group G is a collection T offinite subsets of G , such that: • T ∩ T = ∅ whenever T , T are two different elements of T . • S T ∈T T = G • There exists a finite family S = ( S , . . . , S k ) of finite subsets of G , such that every T can be uniquely represented in the form S j g for some j ∈ { , , . . . , k } and g ∈ G .The elements of T are referred to as tiles , and the elements of S arereferred to as shapes . Also, we can define a mapping σ : T → S suchthat S j = σ ( T ) if and only if T = S j g for some g ∈ G .Every tiling T with shapes S , . . . , S k induces an element x T of { , , . . . , k } G , defined by requesting that if S j g ∈ T for some j , then x T ( g ) = j (the properties constituting the definition of a tiling ensurethat such a j is unique), and if no such j exists, then x T ( g ) = 0. Thisin turn allows us to associate with T a symbolic dynamical system X T defined as the orbit closure of x T under the shift action. Note thatby reversing the procedure which defines x T , we can obtain from everyelement of X T another tiling of G which uses the same collection ofshapes as T .In view of this discussion, it is natural to define the action of G directly on tilings of G by putting: g T = (cid:8) T g − : T ∈ T (cid:9) Clearly, g ( x T )( h ) = j if and only if x T ( hg ) = j , so S j h is a tile of g T if and only if S j hg is a tile of T and this definition is consistent withthe shift action of G on X T .We will need the following result which is an immediate consequenceof theorem 5.2 of [4]: Theorem 4.2. If G is a countable amenable group and K ⊂ G is afinite set, then for every ε > there exists a tiling T of G such that theshapes of T are ( K, ε ) -invariant sets, and the system X T has entropyzero. The main result
We begin by introducing a property analogous to the corner/centregluing conditions used in the Z d context. BARTOSZ FREJ, DAWID HUCZEK
Definition 5.1.
We say that a symbolic dynamical system (
X, G ) hasthe gluing property if there exists a finite set D (containing the neutralelement) such that for any finite subsets T and T of G , such that T ∩ DT = ∅ , and any two blocks A and B , with domains respectively T and T , there exists an x ∈ X such that x ( T ) = A and x ( T ) = B .The set D will be referred to as the gluing distance . Theorem 5.2.
If the symbolic dynamical system ( X, G ) with topologi-cal entropy greater than log k has the gluing property, then there existsa symbolic extension ( e X, G ) of X , having the same topological entropyas X , and such that ( e X, G ) factors onto the full shift over k symbols.Proof. Let l be the number of symbols in the alphabet of X , let γ bea number such that 1 < γ < h ( X )log k , and let D be the gluing distance.There exists a tiling T of G , such that X T has topological entropyzero, and for every shape S of T :(1) | S \ S D | < (( γ −
1) log l k ) | S | .(2) The number of blocks with domain S occurring in X is greaterthan k γ | S | .Indeed, using proposition 3.1 for ε < h ( X ) − γ log k we obtain N and δ such that (2) holds for blocks S which are ( F n , δ )-invariant, where n > N . Then, by theorem 4.2 we get a ( F n , ε )-invariant tiling, where n > N and 0 < ε < δ are appropriately chosen to guarantee (1) (inparticular, we demand that D ⊂ F n ).Combining the properties (1) and (2), we can estimate from belowthe number of blocks with domain S D occurring in X : If we denoteby N the number of blocks with domain S , and by N the number ofblocks with domain S D , then we have k γ | S | < N < N l | S \ S D | < N l (( γ −
1) log l k ) | S | = N k ( γ − | S | , hence N > k | S | . It follows that for every shape S of T we can construct a mapping φ S from the collection of all blocks with domain S D occurring in X onto { , ..., k } S .We can now create the symbolic dynamical system ( e X, G ) as theproduct of (
X, G ) and ( X T , G ). This is obviously an extension of X ,and since the entropy of the product is equal to the sum of entropiesof both systems, e X has entropy equal to X . Every element e x of e X consists of a pair ( x, T ) where x ∈ X and T is a tiling of G using thesame shapes as T . We can now define a map φ : e X → { , ..., k } G asfollows: for every e x = ( x, T ) let y = φ ( e x ) be defined by requiring thatfor every T ∈ T we have y ( T ) = φ σ ( T ) ( x ( T D )). Since y ( T ) dependsonly on x ( T ), this map is continuous. It is also easy to verify that XTENSIONS OF FULL SHIFTS WITH GROUP ACTIONS 5 it commutes with the shift: If e x = ( x, T ) is an element of e X , then g e x = ( gx, g T ). For every tile T g − of g T we have( φ ( g e x ))( T g − ) = φ σ ( T g − ) (cid:0) gx (( T g − ) D ) (cid:1) = φ σ ( T ) ( x ( T D )) = φ ( e x )( T ) , therefore, since T was arbitrary, for every h ∈ G we have( φ ( g e x ))( h ) = φ ( e x )( hg ) . It remains to verify that φ is onto. Let y be any element of { , ..., k } G and let T , T , . . . be an enumeration of the tiles of T and let B i = y ( T i ).There exists some x ∈ X such that φ σ ( T ) ( x (( T ) D )) = B , and thusfor e x = ( x , T ) we have φ ( e x )( T ) = B . Now, suppose that for some j we have already found an e x j ∈ e X such that for every i j we have φ ( e x j )( T i ) = B i (so far we know this is possible for j = 1). There existssome x ′ j +1 ∈ X such that φ σ ( T j +1 ) ( x ′ j +1 (( T j +1 ) D )) = B j +1 . Now, the sets D ( T j +1 ) D and T ∪ T ∪ . . . ∪ T j are disjoint, so the gluing property meansthere exists some x j +1 such that x j +1 ( T i ) = x j ( T i ) for i = 1 , . . . , j , and x j +1 (( T j +1 ) D ) = x ′ j +1 (( T j +1 ) D ). If we now set e x j +1 = ( x j +1 , T ), wewill have φ ( e x j +1 )( T i ) = B i for every i j + 1. By the principleof mathematical induction we obtain that for every j there exists an e x j ∈ e X such that for every i j we have φ ( e x j )( T i ) = B i . Since e X is compact, there exists a convergent subsequence of ( e x j ) convergingto some e x ∈ e X . Hence for every i there exists some j > i such that e x ( T i ) = e x j ( T i ), but then φ ( e x )( T i ) = φ ( e x j )( T i ) = B i . Since i wasarbitrary, φ ( e x ) = y , and thus φ is onto. (cid:3) Acknowledgements
The research is funded by NCN grant 2013/08/A/ST1/00275.
References [1] M. Boyle.
Lower entropy factors of sofic systems , Ergodic Theory Dynam.Systems 3 (1983), no. 4, 541557.[2] M. Boyle, M. Schraudner. Z d shifts of finite type without equal entropy full shiftfactors , Journal of Difference Equations and Applications 15 (2009), 47-52.[3] A. Desai, A class of Z d shifts of finite type which factors onto lowe entropyfull shifts , Proceedings of the American Mathematical Society, Vol. 27, no. 8(2009), 2613-2621[4] T. Downarowicz, D. Huczek, G. Zhang. Tilings of amenable groups , Journalf¨ur die reine und angewandte Mathematik, to appear.[5] A. Johnson, K. Madden.
Factoring higher-dimensional shifts of finite type ontothe full shift , Ergodic Theory and Dynamical Systems 25 (2005), 811-822.[6] E. Lindenstrauss, B. Weiss.,
Mean topological dimension , Israel Journal ofMathematics 115 (2000), no. 1, 1-24.[7] B. Marcus.
Factors and extensions of full shifts , Monatsh. Math. 88 (1979),no. 3, 239247
BARTOSZ FREJ, DAWID HUCZEK
Bartosz Frej, Faculty of Pure and Applied Mathematics, WroclawUniversity of Technology, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wroc law,Poland
E-mail address : [email protected] Dawid Huczek, Faculty of Pure and Applied Mathematics, WroclawUniversity of Technology, Wybrze˙ze Wyspia´nskiego 27, 50-370 Wroc law,Poland
E-mail address ::