On the dynamical asymptotic dimension of a free Z d -action on the Cantor set
aa r X i v : . [ m a t h . OA ] A ug ON THE DYNAMICAL ASYMPTOTIC DIMENSION OF A FREE Z d -ACTION ON THE CANTOR SET ZHUANG NIU AND XIAOKUN ZHOU
Abstract.
Consider an arbitrary extension of a free Z d -action on the Cantor set. It isshown that it has dynamical asymptotic dimension at most 3 d − Introduction
Dynamical Asymptotical Dimension is introduced by Guentner, Willett, and Yu in [2] todescribe the complexity of a topological dynamical system:
Definition 1.1.
Consider a group action X x Γ, where X is a compact Hausdorff spaceand Γ is a discrete group. Its dynamical asymptotic dimension (DAD) is the smallest non-negative integer d such that for any finite subset F ⊆
Γ, there is an open cover U ∪ U ∪· · ·∪ U d of X such that for each U i , 0 ≤ i ≤ d , each x ∈ U i , the cardinality of the set O x := { y ∈ U i : ∃ γ , ..., γ K ∈ F , y = xγ · · · γ K , xγ · · · γ k ∈ U i , ≤ k ≤ K, K ∈ N } is finite and uniformly bounded (with respect to x ).It is shown in [2] that the dynamical asymptotical dimension of any free Z -action is atmost 1, regardless of the space X . It is also shown in [2] that for any discrete group Γ withasymptotic dimension at most d , there is a Γ-action on the Cantor set which has dynamicalasymptotical dimension at most d . In this note, we estimate the dynamical asymptoticaldimension of an arbitrary Z d -action on the Cantor set. In fact, we have the following theorem: Theorem (Theorem 2.8 and Corollary 2.10) . Any extension of a free Z d -action on theCantor set has dynamical asymptotic dimension at most d − . Main result and its proof
Quasi-tilings of Z d . Let us start with certain quasi-tilings (see [3]) of Z d by cubes: Definition 2.1.
Consider Z d . For any natural number l , denote by (cid:3) l the cube (cid:3) l = {− l, − l + 1 , ..., l − , l } d ⊆ Z d . Let r, D, E be natural numbers. An ( r, D, E )-tiling of Z d , denoted by T , is a collection of c i ∈ Z d such that with Dom( T ) = [ i ( c i + (cid:3) D ) , Date : August 11, 2020.
Key words and phrases.
Dynamical asymptotic dimension, free Z d -actions, Cantor system.The research is supported by an NSF grant (DMS-1800882). then,(1) ( c i + (cid:3) D ) ∩ ( c j + (cid:3) D ) = ∅ , i = j ,(2) The (Euclidean) distance between c i + (cid:3) D and c j + (cid:3) D is at least r if i = j , and(3) (cid:3) E ∩ Dom( T ) = ∅ .In other words, an ( r, D, E )-tiling of Z d is a quasi-tiling by cubes of size 2 D + 1, such thattiles are r -separated, but they almost cover 0 up to E .It turns out that if D ≤ E ≤ D , then there are e = 0 , e , e , ..., e d − ∈ Z d such that forany ( r, D, E )-tiling T , one of T , T + e , ..., T + e d − actually covers 0: Lemma 2.2.
For any natural number E , then there are e , e , ..., e s ∈ Z d , where s = 3 d − ,such that if T is an ( r, D, E ) -tiling of Z d for some natural numbers r and D with D ≤ E ≤ D , then ∈ Dom( T ) ∪ Dom( T + e ) ∪ · · · ∪ Dom( T + e s ) , where s = 3 d − .Proof. Set { e , e , ..., e d − } = { ( n , n , ..., n d ) ∈ Z d : n i ∈ { , ± E }} , with e = (0 , ..., / ∈ Dom( T ),then, at least one of e i , i = 1 , ..., d − , is in Dom( T ).Assume none of e i was inside Dom( T ). Then one asserts that (cid:3) E ∩ Dom( T ) = ∅ . This contradicts Condition (3) and hence proves the lemma.For the assertion, assume there is c ∈ Z d with c + (cid:3) D ⊆ Dom( T ) and (cid:3) E ∩ ( c + (cid:3) D ) = ∅ . Then there exist − E ≤ n i ≤ E, ≤ i ≤ d, such that ( n , ..., n d ) ∈ c + (cid:3) D . Note that (cid:3) E ∩ ( c + (cid:3) D ) = ∅ implies − D − E ≤ c i ≤ D + E, ≤ i ≤ d, c = ( c , c , ..., c d );and also note c + (cid:3) D = { ( c + s , c + s , ..., c d + s d ) : − D ≤ s i ≤ D } . For each c i , if | c i | ≥ E , then choose s i ∈ [ − D, D ] such that | c i + s i | = E ; if | c i | ≤ D , thenchoose s i = − c i so that c i + s i = 0; if D ≤ | c i | ≤ E , then choose s i ∈ [ − D, D ] such that | c i + s i | = E (note that one assumes E ≤ D ). With this choice of s i , one has that c + (cid:3) D contains at least one of e i , and so such e i is inside Dom( T ). This contradicts the assumption,and proves the assertion. (cid:3) YNAMICAL ASYMPTOTIC DIMENSION 3
Group actions and equivariant quasi-tilings.
Recall
Definition 2.3.
Let X be a topological space and let Γ be a discrete group. By a (right)Γ-action on X , denoted by X x Γ, we mean a continuous map X × Γ ∋ ( x, γ ) → xγ ∈ X such that xe = x and ( xγ ) γ = x ( γ γ ) , x ∈ X, γ γ ∈ Γ . We say a Γ-action on X is free if xγ = x for some x ∈ X and γ ∈ Γ implies γ = e .Consider actions X x Γ and Y x Γ. We say that X x Γ is an extension of Y x Γ (or Y x Γ is a factor of X x Γ) if there is a quotient map π : X → Y such that π ( xγ ) = π ( x ) γ, x ∈ X, γ ∈ Γ . Definition 2.4.
Consider an Z d -action on topological space X . A set-valued map X ∋ x
7→ T ( x ) ∈ Z d is said to be equivariant if T ( xn ) = T ( x ) − n, where T ( x ) − n is the translation of T ( x ) by − n .The map x
7→ T ( x ) is said to be continuous if for any R > x ∈ X , there is anopen set U ∋ x such that T ( y ) ∩ B R = T ( x ) ∩ B R , y ∈ U, where B R is the ball in Z d with center 0 and radius R . Lemma 2.5.
Consider an Z d -action on a topological space X . Let N ∈ N , and let x
7→ T ( x ) be a continuous equivariant map with value ( r, D, E ) -tilings of Z d with r > N √ d . Put Ω = { x ∈ X : 0 ∈ Dom( T ( x )) } . Then, Ω is open. Moreover, for any x ∈ X , one has (cid:12)(cid:12) { n ∈ Z d : n = n + · · · + n K , x ( n + · · · + n k ) ∈ Ω , k n k k ∞ ≤ N, (2.1) 1 ≤ k ≤ K, K ∈ N }|≤ (2 D + 1) d . Proof.
The openness of Ω follows directly from the continuity of the map x
7→ T ( x ). Let usshow the estimate (2.1).Pick x ∈ Ω, and write c + (cid:3) D to be the tile of T ( x ) containing 0. Since the function x
7→ T ( x ) is equivariant, one has that T ( xn ) = T ( x ) − n ; hence, by Condition (2), for any n ∈ Z d with k n k ∞ ≤ N , one has that either 0 is in the tile c + (cid:3) D − n (therefore x n ∈ Ωand c − n ∈ (cid:3) D ) or 0 / ∈ Dom( T ( x n )) (therefore x n / ∈ Ω).Thus, if there are n , n , ..., n K ∈ Z d with k n k k ∞ ≤ N and n x ∈ Ω , x ( n + n ) ∈ Ω , ..., x ( n + · · · + n K ) ∈ Ω , ZHUANG NIU AND XIAOKUN ZHOU one has c − n ∈ (cid:3) D , c − n − n ∈ (cid:3) D , ..., c − n − · · · − n K ∈ (cid:3) D , and hence n = n + · · · + n K ∈ c + (cid:3) D . Since | c + (cid:3) D | = | (cid:3) D | = (2 D + 1) d , this proves the lemma. (cid:3) Cantor systems and an estimate of dynamical asymptotic dimension.
Let usfocus on extensions of a free Z d -action on the Cantor set, which is the unique compactseparable Hausdorff space that is totally disconnected and perfect.First, for any free Z d -action on the Cantor set, equivariant continuous ( r, D, E )-tiling-valued functions always exist: Proposition 2.6.
Consider a free Z d -action on X where X is the Cantor set, and let N ∈ N be arbitrary. Then, there are natural numbers r, D, E with r > N √ d and D ≤ E ≤ D , anda continuous equivariant map x
7→ T ( x ) on X such that each T ( x ) a ( r, D, E ) -tiling of Z d .Proof. The construction is similar to that of Lemma 3.4 of [1].Pick a natural number r > N √ d , and then pick a natural number L > r . Since theaction is free and X is the Cantor set, by a compactness argument, one obtains mutuallydisjoint clopen sets U , U , ..., U s , such that X = U ∪ U ∪ · · · ∪ U s , and for each U i , 1 ≤ i ≤ s , the open sets U i n, n ∈ (cid:3) L , are mutually disjoint.Start with U . For each x ∈ X , put C ( x ) = { n ∈ Z d : xn ∈ U } , · · · · · · · · ·C i ( x ) = C i − ( x ) ∪ { n ∈ Z d : xn ∈ U i , ( n + (cid:3) L ) ∩ ( C i − ( x ) + (cid:3) L ) = ∅ } , · · · · · · · · ·C s ( x ) = C s − ( x ) ∪ { n ∈ Z d : xn ∈ U s , ( n + (cid:3) L ) ∩ ( C s − ( x ) + (cid:3) L ) = ∅ } . Since U is clopen, the map x
7→ C ( x ) is continuous in the sense that for any x and any R >
0, there is a neighbourhood W of x such that C ( y ) ∩ B R = C ( x ) ∩ B R , y ∈ W. Consider the map x
7→ C ( x ). Fix x ∈ X , R >
0. Since U is clopen, there is a neighbour-hood W of x such that { n ∈ Z d : xn ∈ U } ∩ B R = { n ∈ Z d : yn ∈ U } ∩ B R , y ∈ W. Note that x
7→ C ( x ) is continuous, then the neighbourhood W can be chosen so that( C ( x ) + (cid:3) L ) ∩ B R = ( C ( x ) + (cid:3) L ) ∩ B R , y ∈ W, YNAMICAL ASYMPTOTIC DIMENSION 5 and therefore for any y ∈ W , { xn ∈ U , ( n + (cid:3) L ) ∩ ( C ( x ) + (cid:3) L ) = ∅ } ∩ B R = { yn ∈ U , ( n + (cid:3) L ) ∩ ( C ( y ) + (cid:3) L ) = ∅ } ∩ B R . Together with the continuity of x
7→ C ( x ), this shows that x
7→ C ( x ) is continuous.Repeat this argument, one shows that the map x
7→ C s ( x ) is continuous.Let us show that the map x
7→ C s ( x ) is equivariant. Start with x
7→ C ( x ). Let n ∈ Z d and consider xn . Since xm ∈ U if and only if x ( n + m − n ) ∈ U , one has C ( xn ) = C ( x ) − n. A similar argument shows that C ( x ) , ..., C s ( x ) are equivariant.One asserts that ( c + (cid:3) L ) ∩ ( c + (cid:3) L ) = ∅ , c = c , c , c ∈ C s ( x ) . Indeed, since U n , n ∈ (cid:3) L , are mutually disjoint, one has that( c + (cid:3) L ) ∩ C ( x ) = c, c ∈ C ( x ) , and thus ( c + (cid:3) L ) ∩ ( c + (cid:3) L ) = ∅ , c = c , c , c ∈ C ( x ) . Now, pick c , c ∈ C ( x ) = C ( x ) ∪ { n ∈ Z d : xn ∈ U , ( n + (cid:3) L ) ∩ ( C ( x ) + (cid:3) L ) = ∅ } . If c , c ∈ C ( x ), then as shown above,( c + (cid:3) L ) ∩ ( c + (cid:3) L ) = ∅ . Assume that c , c ∈ { n ∈ Z d : xn ∈ U , ( n + (cid:3) L ) ∩ ( C ( x ) + (cid:3) L ) = ∅ } ⊆ { n ∈ Z d : xn ∈ U } . Then, since U n , n ∈ (cid:3) L , are mutually disjoint, the same argument as that of C ( x ) showsthat ( c + (cid:3) L ) ∩ ( c + (cid:3) L ) = ∅ . Assume that c ∈ C and c ∈ { n ∈ Z d : xn ∈ U , ( n + (cid:3) L ) ∩ ( C ( x ) + (cid:3) L ) = ∅ } . Then theequation ( c + (cid:3) L ) ∩ ( c + (cid:3) L ) = ∅ just follows from the definition.Repeat this argument for C ( x ) , ..., C s ( x ), and this proves the assertion.Note that for the given x , there exists a U i containing x . Therefore, either (cid:3) L ∩ ( C i − ( x ) + (cid:3) L ) = ∅ or 0 ∈ C i ( x ) . In particular, one always has that (cid:3) L ∩ ( C i ( x ) + (cid:3) L ) = ∅ , and hence (cid:3) L ∩ ( C s ( x ) + (cid:3) L ) = ∅ . To summarize, setting C ( x ) = C s ( x ), one obtains a continuous equivariant map x
7→ C ( x )satisfying ZHUANG NIU AND XIAOKUN ZHOU (1) ( c i + (cid:3) L ) ∩ ( c j + (cid:3) L ) = ∅ , c i = c j , c i , c j ∈ C s ( x ) and(2) (cid:3) L ∩ ( C s ( x ) + (cid:3) L ) = ∅ ;hence it satisfies(3) ( c i + (cid:3) L − r ) ∩ ( c j + (cid:3) L − r ) = ∅ , c i = c j , c i , c j ∈ C s ( x ),(4) (cid:3) L + r ∩ ( C s ( x ) + (cid:3) L − r ) = ∅ ;and, moreover(5) the (Euclidean) distance between c i + (cid:3) L − r and c j + (cid:3) L − r is at least r if c i = c j .Thus, each C ( x ) is an ( r, L − r, L + r ) tiling. Since L > r , one has L + r < L − r ), andthis proves the statement of the proposition. (cid:3) Corollary 2.7.
Consider a free Z d -action on X where X is the Cantor set, and let N ∈ N be arbitrary. Then, there exist continuous equivariant maps x
7→ T i ( x ) , i = 0 , , ..., d − , with each T i ( x ) a ( r, D, E ) -tilings of Z d for some r, D, E ∈ N with r > N √ d , such that, ifput Ω i = { x ∈ X : 0 ∈ Dom( T i ( x )) } , i = 0 , , ..., d − , then Ω ∪ Ω ∪ · · · ∪ Ω d − = X. Proof.
It follows from Proposition 2.6 that there are natural numbers r, D, E with r > N √ d and D ≤ E ≤ D, and a continuous equivariant map x
7→ T ( x ) on X such that each T ( x ) a ( r, D, E )-tiling of Z d .Consider the translations of the function T : T = T + e , T = T + e , ..., T d − = T + e d − , where e , ..., e d − are the vectors (with repect to E ) obtained from Lemma 2.2. Since D ≤ E ≤ D , it follows from Lemma 2.2 that for any x ∈ X , one has0 ∈ Dom( T ( x )) ∪ Dom( T ( x )) ∪ · · · ∪ Dom( T d − ( x )) , and thus Ω ∪ Ω ∪ · · · ∪ Ω d − = X, as desired. (cid:3) Theorem 2.8.
The dynamical asymptotic dimension of any free Z d -action on the Cantorset is at most d − .Proof. Let N ∈ N be arbitrary. It follows from Corollary 2.7 that there exist continuousequivariant maps x
7→ T i ( x ) , i = 0 , , ..., d − , with each T i ( x ) a ( r, D, E )-tilings of Z d for some r, D, E with r > N √ d withΩ ∪ Ω ∪ · · · ∪ Ω d − = X, YNAMICAL ASYMPTOTIC DIMENSION 7 where Ω i = { x ∈ X : 0 ∈ Dom( T i ( x )) } , i = 0 , , ..., d − , which is open.Since r > N √ d , by Lemma 2.5, for any i = 0 , , ..., d , one has (cid:12)(cid:12) n ∈ Z d : n = n + · · · + n K , x ( n + · · · + n k ) ∈ Ω i , k n k k ∞ ≤ N, ≤ k ≤ K, K ∈ N }|≤ (2 D + 1) d < + ∞ . That is, the dynamical asymptotic dimension of X x Z d is at most 3 d − (cid:3) Lemma 2.9.
Let X x Γ be an extension of a free action Y x Γ . Then the dynamicalasymptotic dimension of X x Γ is at most the dynamical asymptotic dimension of Y x Γ .Proof. Let d ∈ Z such that the dynamical asymptotical dimension of Y x Γ is at most d .Let Γ ⊆ Γ be finite. Then, together with the freeness of Y x Γ, there exist an open cover U ∪ U ∪ · · · ∪ U d of Y and M > U i , 0 ≤ i ≤ d , y ∈ U i , one has that(2.2) |{ γ · · · γ K : ∃ γ , ..., γ K ∈ Γ , y γ · · · γ k ∈ U i , ≤ k ≤ K, K ∈ N }| ≤ M. Consider the open sets π − ( U ) , π − ( U ) , ..., π − ( U d ) , where π : X → Y is the quotient map, and note that they form an open cover of X . Foreach 0 ≤ i ≤ d , pick an arbitrary x ∈ π − ( U i ) and assume there are γ , ..., γ K ∈ Γ for some K ∈ N such that x ∈ π − ( U i ) , x γ ∈ π − ( U i ) , ..., x γ γ · · · γ K ∈ π − ( U i ) . Applying the quotient map π , one has π ( x ) ∈ U i , π ( x ) γ ∈ U i , ..., π ( x ) γ γ · · · γ K ∈ U i , and, by (2.2), this implies (cid:12)(cid:12) { γ · · · γ K : ∃ γ , ..., γ K ∈ Γ , x γ · · · γ k ∈ π − ( U i ) , ≤ k ≤ K, K ∈ N } (cid:12)(cid:12) ≤ M. Thus, the dynamical asymptotic dimension of X x Γ is at most d . (cid:3) Then, the following is a straightforward corollary of Theorem 2.8:
Corollary 2.10.
The dynamical asymptotic dimension of any extension of a free Z d -actionon the Cantor set is at most d − . References [1] T. Downarowicz and D. Huczek. Dynamical quasitilings of amenable groups.
Bull. Pol. Acad. Sci.Math. , 66(1):45–55, 2018. URL: https://mathscinet.ams.org/mathscinet-getitem?mr=3782587 , doi:10.4064/ba8128-1-2018 .[2] E. Guentner, R. Willett, and G. Yu. Dynamic asymptotic dimension: relation todynamics, topology, coarse geometry, and C ∗ -algebras. Math. Ann. , 367(1-2):785–829, 2017. URL: https://mathscinet.ams.org/mathscinet-getitem?mr=3606454 , doi:10.1007/s00208-016-1395-0 . ZHUANG NIU AND XIAOKUN ZHOU [3] D. S. Ornstein and B. Weiss. Entropy and isomorphism theorems for actions of amenable groups.
J.Analyse Math. , 48:1–141, 1987. URL: https://mathscinet.ams.org/mathscinet-getitem?mr=910005 , doi:10.1007/BF02790325 . Department of Mathematics, University of Wyoming, Laramie, Wyoming, USA, 82071
E-mail address : [email protected] Department of Mathematics, University of Wyoming, Laramie, Wyoming, USA, 82071
E-mail address ::