aa r X i v : . [ m a t h . D S ] F e b FINITE MEAN DIMESNION AND MARKERPROPERTY
RUXI SHI
Abstract.
In this paper, we develop the theory of Z p -index whichhas been introduced by Tsukamoto, Tsutaya and Yoshinaga. As anapplication, we show that given any positive number, there existsa dynamical system with mean dimension equal to such numbersuch that it does not have the marker property. Introduction A (topological) dynamical system is a pair ( X, T ) where X is compactmetrizable space and T : X → X is a homeomorphism. The most basicinvariant of dynamical systems is topological entropy which has beenstudied for a long time. Gromov [Gro99] introduced a new topologi-cal invariant of dynamical systems called mean topological dimension (see the precise definition in Section 2). It was further developed byLindenstrauss and Weiss [LW00]. Roughly speaking, mean topologi-cal dimension captures the complexity of dynamical systems of infiniteentropy.For a dynamical system ( X, T ), let P n ( X, T ) be the set of n -periodicpoints, i.e. P n ( X, T ) := { x ∈ X : T n x = x } . The dynamical system (
X, T ) is said to be aperiodic if P n ( X, T ) = ∅ forall n ≥
1. For a dynamical system without fixed points (i.e. P = ∅ ),Tsukamoto, Tsutaya and Yoshinaga [TTY20] introduced Z p -index tostudy the set of p -periodic points for prime number p . They showed itsgrowth is at most linear in p . Moreover, they applied the theory of Z p -index to study the marker property. For a dynamical system ( X, T ),it is said to satisfy the marker property if for each positive integer N there exists an open set U ⊂ X satisfying that U ∩ T − n U = ∅ for 0 < n < N and X = ∪ n ∈ Z T n U. For example, an extension of an aperiodic minimal system has themarker property [Lin99, Lemma 3.3]. Obviously, the marker propertyimplies the aperiodicity.The marker property has been extensively used in the theory of meandimension. Lindenstrauss [Lin99] proved that for a dynamical system
Mathematics Subject Classification.
Primary: 37B05, 55M35.
Key words and phrases. Z p -index, mean dimension, marker property. having the marker property, it has zero mean dimension if and only ifit is isomorphic to an inverse limit of finite entropy dynamical systems.Gutman, Qiao and Tsukamoto [GQT19] showed that a dynamical sys-tem having the marker property with mean dimension smaller than N/ , N ) Z . Linden-strauss and Tsukamoto [LT19] proved that if a dynamical system hasthe marker property, then one can find a metric such that the uppermetric mean dimension is equal to mean dimension of this dynamicalsystem. The metric mean dimension was introduced by Lindenstraussand Weiss [LW00] which majors the value of mean topological dimen-sion. See also [Gut15, GLT16, Gut17, GT20, Tsu20] and referencestherein where the marker property was investigated.As we mentioned before, the marker property implies the aperiodic-ity. For several years, it has been specially curious whether the converseis true, i.e. whether the aperiodicity implies the marker property. Thisopen problem was stated by Gutman ([Gut15, Problem 5.4], [Gut17,Problem 3.4]) and its origin goes back to the paper of Lindenstrauss[Lin99]. Very recently, Tsukamoto, Tsutaya and Yoshinaga [TTY20]solved this problem by constructing a counter-example of an aperiodicdynamical system with infinite mean dimension which does not havethe marker property. It is very surprising that such a counter-examplecomes from an application of Z p -index theory which they studied in thesame paper. Since the example that they constructed is infinite meandimensional, they proposed the following problem [TTY20, Problem7.6]. Problem 1.1.
Construct an aperiodic dynamical system with finitemean dimension which does not have the marker property.
As Tsukamoto, Tsutaya and Yoshinaga commented in their paper,they believed that such an example exists but it seems difficult toconstruct such example by their current method.In this paper, we continue to investigate Z p -index theory and its con-nection to dynamical systems. As an application, we give an affirmativeanswer to Problem 1.1. Furthermore, we show that the desired dynam-ical system that we constructed could have mean dimension arbitrarilyclose to zero. Our main result is stated as follows. Theorem 1.2.
For any η > , there exists an aperiodic dynamicalsystem with mean dimension small than η which does not have themarker property. Now we present the main idea and explain what are the main diffi-culties in the proof Theorem 1.2. There are two principal steps in theproof.Step 1. We first find a family of dynamical systems with same pe-riodic coindex (see the precise definition in Section 3) and uniformly
INITE MEAN DIMESNION AND MARKER PROPERTY 3 bounded mean dimension from above which has the universal prop-erty (see the discussion in Section 4). In fact, these dynamical systemsturn out to be subshifts of (( R / Z ) N ) Z for some positive number N .We choose a sequence of dynamical systems ( X n , T n ) in such familysatisfying that the smallest period is getting larger and larger as n tends to infinity. Then we show that the inverse limit of { ( X n , T n ) } n ∈ N is an aperiodic dynamical system with finite mean dimension whichdoes not have the marker property (Theorem 5.1). This gives a posi-tive answer to Problem 1.1. The main difficulties that we face in theabove construction is the following two issues. The first one is that weneed certain group structure in the base space of subshifts. This is thereason why we can not apply directly the universal dynamical systemsstudied in [TTY20] to our proof. The second one is that we need a con-tinuous left inverse map of the nature projection of the inverse limit(see the defintion of γ m in Section 5). The disadvantage of using theinverse limit is that we can not expect the result (Lemma 5.7) similarto [TTY20, Lemma 6.1] holding for all inverse limits (unlike [TTY20,Lemma 6.1] holds for all infinite product). However, the advantageis that the mean dimension of the inverse limit is controlled by thesuperior of the mean dimensions of ( X n , T n ) (See Proposition 5.8).Step 2. Based on the dynamical system ( X, T ) (i.e. the inverse limitof { ( X n , T n ) } n ∈ N constructed in Step 1), we build a n -time system( X × Z n , T n ) (see the precise definition in Section 6) such that it hasthe marker property for every n ∈ N . Since the mean dimension of n -time system ( X × Z n , T n ) is the mean dimension of ( X, T ) dividedby n , we get the desired result.By Theorem 1.2 (and as well as Corollary 7.1), one can not expectthat the marker property implies aperiodicity for dynamical systemswith positive mean dimension. The zero mean dimensional ones isthe only case left. It seems to be believable that the marker propertyand zero mean dimension together imply aperiodicity. In fact, this isequivalent to a conjecture of Lindenstrauss [Lin99]. See the discussionin Section 7.The paper is organized as follows. In Section 2, we recall the def-initions of inverse limit and mean dimension. In Section 3, we recallthe Z p -coindex of dynamical system which was introduced in [TTY20].In Section 4, we construct a universal dynamical system without fixedpoints. In Section 5, we construct an aperiodic dynamical system withfinite mean dimension which does not have marker property. In Sec-tion 6, we construct an aperiodic dynamical system without markerproperty whose mean dimension is arbitrarily small. In Section 7, wediscuss open problems and conjectures. RUXI SHI Preliminaries
For dynamical systems (
X, T ) and (
Y, S ), a map f : X → Y is saidto be equivariant if f ◦ T = S ◦ f . A dynamical system ( X, T ) is calleda topological factor of a system (
Y, S ) if there exists a continuous andequivariant surjection π : X → Y . The map π is called a factor map .2.1. Inverse limit of dynamical systems.
Let { ( X n , T n ) } n ∈ N be asequence of dynamical systems. Suppose for every pair m > n thereexists a factor map σ m,n : X m → X n such that for any triple m > n > l the diagram ( X m , T m ) σ m,n (cid:15) (cid:15) σ m,l x x rrrrrrrrrr ( X l , T l ) ( X n , T n ) σ n,l o o commutes. Set X = { ( x n ) n ∈ N ∈ Y n ∈ N X n : σ m,n ( x m ) = x n , ∀ m > n } . Clearly, the space X is closed in Q n ∈ N X n and thus compact. Let π n : X → X n be the natural projection map for each n ∈ N . Definethe continuous equivariant map T : X → X by T : ( x n ) n ∈ N → ( T n x n ) n ∈ N . It satisfies that σ m,n ◦ π m = π n for all m > n . We call the dynamicalsystem ( X, T ) the inverse limit of the family { ( X n , T n ) } n ∈ N via σ =( σ m,n ) m,n ∈ N ,m>n and write( X, T ) = lim ←− ( X n , T n ) . Mean dimension.
Let X be a compact space. For two finiteopen covers A and B of X , we say that the cover B is finer than thecover A , and write B ≻ A , if for every element of B , one can find anelement of A which contains it. For a finite open cover A of X , wedefine the quantities ord( A ) := sup x ∈ X X A ∈A A ( x ) − , and D ( A ) := min B≻A ord( B ) . Clearly, if
B ≻ A then D ( B ) ≥ D ( A ). The (topological) dimension of X is defined by dim( X ) := sup A D ( A ) , where A runs over all finite open covers of X . For finite open covers A and B , we set A ∨ B := { U ∩ V : U ∈ A , V ∈ B} . It is easy to checkthat D ( A ∨ B ) ≤ D ( A ) + D ( B ). INITE MEAN DIMESNION AND MARKER PROPERTY 5
Let (
X, T ) be a dynamical system. The mean dimension of (
X, T )is defined by mdim(
X, T ) = sup α lim n →∞ D ( ∨ n − i =0 T − i α ) n , where A runs over all finite open covers of X . The limit above existsis due to the sub-additivity of D .We mention some basic properties of D and mean dimension. Referto the book [Coo05] for the proofs and further properties. • Let X and Y be topological spaces and let f : X → Y bea continuous map. Let A be a finite open cover of Y . Then D ( f − ( A )) ≤ D ( A ). • If (
Y, T ) is a subsystem of (
X, T ), i.e. Y is a closed T -invariantsubspace of X , then mdim( Y, T ) ≤ mdim( X, T ). • mdim( X, T n ) = n · mdim( X, T ).3.
Coindex of free Z p system We denote by Z p := Z /p Z for prime number p through this paper.A pair ( X, T ) is called a Z p -system if ( X, T ) is a dynamical system and T induces a Z p -action on X . Moreover, the Z p -system is said to be free if it has no fixed points, i.e. T x = x for all x ∈ X . Since p is prime,a Z p -system is free if and only if it is aperiodic, i.e. T n x = x for all1 ≤ n ≤ p − x ∈ X .A free Z p -system ( X, T ) is called an E n Z p -system if it satisfies thefollowing conditions: • X is an n -dimensional finite simplicial complex. • T : X → X is a simplical map, i.e. it maps every simplex to asimplex affinely. • X is ( n − k -th homotopy group π k ( X ) = 0for all k ≤ n − Z p acts freely by T on a finite set Y , then ( Y, T ) isan E Z p -system. Moreover, if Y i are finite sets and ( Y i , T i ) are free Z p -systems for 0 ≤ i ≤ n , then ( Y ∗ Y ∗ · · · ∗ Y n , T ∗ T ∗ · · · ∗ T n ) isan E n Z p -system. Recall that for dynamical systems ( X, T ) and (
Y, S ),the dynamical systems ( X ∗ Y, T ∗ S ) is defined by X ∗ Y := [0 , × X × Y / ∼ and T ∗ S ( t, x, y ) = ( t, T x, Sy )where the equivalence relation ∼ is given by(0 , x, y ) ∼ (0 , x, y ′ ) and (1 , x, y ) ∼ (1 , x ′ , y ) , for any x, x ′ ∈ X and any y, y ′ ∈ Y . It is easy to see that P n ( X ∗ Y, T ∗ S ) = P n ( X, T ) ∗ P n ( Y, S ) , ∀ n ∈ N . In particular, if (
X, T ) and (
Y, S ) are both aperiodic, then so is ( X ∗ Y, T ∗ S ). RUXI SHI
Due to [Mat03, Lemma 6.2.2], there is an equivariant and continuousmap from one E n Z p -system to another. Thus, the E n Z p -system is“unique” in the sense that we regard two systems as the same wheneverthere are equivariant continuous maps between each other.Following [TTY20], we define the coindex of a free Z p -system ( X, T )bycoind p ( X, T ) := max { n ≥ ∃ an equivariant continuous f : E n Z p → X } We use the convention that coind p ( X, T ) = − X = ∅ . Notice thatif there exists an equivariant continuous map from E m Z p to E n Z p then m ≤ n (see [Mat03, Theorem 6.2.5]). Thuscoind p ( E n Z p , T ) = n. Moreover, if (
X, T ) is an E n Z p -system, then coind p ( X, T ) = n . Lemma 3.1.
Let ( X, T ) be a free Z p -system. For any integer n coprimewith p , we have coind p ( X, T n ) = coind p ( X, T ) . Proof.
It follows directly from the fact that for any integer n coprimewith p if ( Y, S ) is an E n Z p -system then ( Y, S n ) is not only an E n Z p -system but also topologically conjugate to ( Y, S ). (cid:3) The following basic properties of coindex were proved by Tsukamoto,Tsutaya and Yoshinaga.
Proposition 3.2 ([TTY20], Proposition 3.1) . Let ( X, T ) and ( Y, S ) be free Z p -systems. Then the following properties hold. (1) If there is an equivariant continuous map f : X → Y then coind p ( X, T ) ≤ coind p ( Y, S ) . (2) The system ( X ∗ Y, T ∗ S ) is a free Z p -systems and coind p ( X ∗ Y, T ∗ S ) ≥ coind p ( X, T ) + coind p ( Y, S ) + 1 . Let (
X, T ) be a dynamical system (
X, T ) without fixed points (i.e.
T x = x for all x ∈ X ). Recall that P n ( X, T ) is the set of n -periodicpoints. Observe that ( P p ( X, T ) , T ) is a free Z p -system for prime num-ber p . We define the periodic coindex of ( X, T ) ascoind
Per p ( X, T ) = coind p (( P p ( X, T )) , T ) , for prime numbers p . Corollary 3.3.
Let ( X, T ) and ( Y, S ) be dynamical systems withoutfixed points. Let p be a prime number. Then the following propertieshold. (1) If there is an equivariant continuous map f : X → Y then coind Per p ( X, T ) ≤ coind Per p ( Y, S ) . (2) The system ( X ∗ Y, T ∗ S ) has no fixed points and coind Per p ( X ∗ Y, T ∗ S ) ≥ coind Per p ( X, T ) + coind
Per p ( Y, S ) + 1 . INITE MEAN DIMESNION AND MARKER PROPERTY 7
Proof. (1) Notice that f restricted to the set of p -periodic points in-duces an equivariant continuous map from ( P p ( X, T ) , T ) to ( P p ( Y, S ) , S ).Then we apply Proposition 3.2 (1).(2) Observe the fact that P p ( X ∗ Y, T ∗ S ) = P p ( X, T ) ∗ P p ( Y, S ).Then we apply Proposition 3.2 (2). (cid:3)
It follows from Corollary 3.3 (1) that the sequence (coind
Per p ( · )) p is atopological invariant (i.e. it is invariant under topological conjugacy)in the catalogue of dynamical systems without fixed points. Moreover,if we use the convention that coind Per p ( · ) = − Per p ( · )) p becomes a topological invariant ofdynamical systems.4. Universal dynamical system without fixed points
Let S = R / Z . In what follows, we also regard S as the interval[ − ,
1] such that the endpoints − ρ be the metric on S defined by ρ ( x, y ) = min n ∈ Z | x − y − n | . It is easy to see that the diameter of S under the metric ρ is 1 and ρ is homogeneous , i.e. ρ ( x + z, y + z ) = ρ ( x, y ).Let N be a positive integer. Let ρ N be the metric on S N defined by ρ N ( x, y ) = max ≤ i ≤ N ρ ( x i , y i ) , where x = ( x i ) ≤ i ≤ N , y = ( y i ) ≤ i ≤ N ∈ S N . Then the diameter of S N under the metric ρ N is 1. Notice that S N is an abelian group and ρ N ( x + z, y + z ) = ρ N ( x, y ) for any x, y, z ∈ S N (because ρ ( x i + z i , y i + z i ) = ρ ( x i , y i ) as we have mentioned before). The product space ( S N ) Z is compact and metrizable under the product topology and the metric d ( x, y ) = P n ∈ Z n ρ N ( x n , y n ) where x = ( x n ) n ∈ N and y = ( y n ) n ∈ N . Let σ : ( S N ) Z → ( S N ) Z be the shift map, i.e. σ : ( x n ) n ∈ Z → ( x n +1 ) n ∈ Z . Inthis paper, we always use σ to denote the shift map on ( S N ) Z and aswell as its subshift.Tsukamoto, Tsutaya and Yoshinaga [TTY20] studied the universalproperty of certain subsystem in Hilbert cube ([0 , N ) Z . Notice that[0 ,
1] can be embedded into S by x x ; conversely, S can be embed-ded into [0 , = { x + yi : x, y ∈ [0 , } by x e πix + e + i . It isreasonable to expect that some similar universal property is valid forcertain subshifts of ( S N ) Z . We investigate such universal property inthis section.A continuous map f from a metric space ( X, d ) to a space Y is calledan ǫ -embedding if f ( x ) = f ( y ) implies d ( x, y ) < ǫ . Lemma 4.1.
Let ǫ > . Let ( X, d ) be a compact metric space. Thereexists an integer N > and an ǫ -embedding from X to S N . RUXI SHI
Proof.
Without loss of generality, we assume diam( X ) ≤ /
4. Pickan open cover { B ( x i , ǫ ) } ≤ i ≤ N of X where B ( x, ǫ ) is the open ballcentered at x of radius ǫ . Define a continuous map f : X → S N by f ( x ) = ( d ( x, x ) , d ( x, x ) , . . . , d ( x, x N )) . For x, y ∈ X with f ( x ) = f ( y ), supposing x ∈ B ( x i , ǫ ), we obtain that d ( y, x i ) = d ( x, x i ) < ǫ/ d ( x, y ) < ǫ . This implies that f is ǫ -embedding. (cid:3) For any positive integers m, N and any number δ >
0, we define asubsystem ( X ( N, m, δ ) , σ ) of (( S N ) Z , σ ) by X ( N, m, δ ) := { ( x n ) n ∈ Z ∈ ( S N ) Z : ρ N ( x n , x n + m ) ≥ δ, ∀ n ∈ Z } . It is clear that ( X ( N, m, δ ) , σ ) has no fixed points. The universal prop-erty of X ( N, , δ ) is studied in the following lemma. Lemma 4.2.
Let ( X, T ) be a dynamical system without fixed points.Then there exists an positive integer N , an δ > and an equivariantcontinuous map from X to X ( N, , δ ) .Proof. Let d be a metric on X which is compatible with its topology.Since ( X, T ) has no fixed point, we have inf x ∈ X d ( x, T x ) >
0. Pick0 < ǫ < inf x ∈ X d ( x, T x ). By Lemma 4.1, there is an integer N and an ǫ -embedding f from X to S N . Define an equivariant and continuousmap g : X → ( S N ) Z by x ( f ( T n x )) n ∈ Z . Since X is compact and f is a ǫ -embedding, there is an δ > | f ( x ) − f ( y ) | < δ implies d ( x, y ) < ǫ . Then we get that | f ( x ) − f ( T x ) | ≥ δ for all x ∈ X . It follows that the image of X under g is contained in X ( N, , δ ). This completes the proof. (cid:3) Even though we can not replace X ( N, , δ ) in Lemma 4.2 by X ( N, m, δ )for m ≥
2, the following lemma tells us that X ( N, m, δ ) is the same as X ( N, , δ ) in terms of periodic coindex for large prime numbers. Lemma 4.3.
Let < δ < . For any positive integer m and any primenumber p > m , we have coind Per p ( X ( N, m, δ ) , σ ) = coind Per p ( X ( N, , δ ) , σ ) ≥ . Proof.
Fix a positive integer m and a prime number p > m . Noticethat the set of p -periodic points P p ( X ( N, m, δ )) = { ( x i ) i ∈ Z p ∈ ( S N ) Z p : ρ N ( x n , x n + m ) ≥ δ, ∀ n ∈ Z p } , INITE MEAN DIMESNION AND MARKER PROPERTY 9 which is not empty. Define f j : ( S N ) Z p → ( S N ) Z p by ( x i ) i ∈ Z p ( x ij ) i ∈ Z p . Let 1 ≤ k ≤ p − m in Z p , i.e. km ≡ p . Then the following diagram commutes: P p ( X ( N, m, δ ) , σ ) σ m / / f m (cid:15) (cid:15) P p ( X ( N, m, δ ) , σ ) f k (cid:15) (cid:15) P p ( X ( N, , δ ) , σ ) σ / / f k O O P p ( X ( N, , δ ) , σ ) f m O O It means that the system ( P p ( X ( N, m, δ ) , σ ) , σ m ) is topologically con-jugate to ( P p ( X ( N, , δ ) , σ ) , σ ). Then by Lemma 3.1, we obtain thatcoind Per p ( X ( N, , δ ) , σ ) = coind p ( P p ( X ( N, m, δ ) , σ ) , σ m )= coind p ( P p ( X ( N, m, δ ) , σ ) , σ )= coind Per p ( X ( N, m, δ ) , σ ) . This completes the proof. (cid:3)
Lemma 4.4.
There exists a dynamical system ( Y, S ) satisfying that P ( Y, S ) = ∅ and < ♯ ( P n ( Y, S )) < + ∞ for n ≥ . Moreover, we have coind Per p ( Y, S ) = 0 for all prime number p .Proof. Let Y := { ( x n ) n ∈ N ∈ { , } Z : x n x n +1 x n +2 = 000 or 111 } . Let S : Y → Y be the shift. Then it is easy to check that ( Y, S ) has nofixed points and 0 < ♯P n ( Y, S ) < + ∞ for n ≥
2. Since P n ( Y, S ) is non-empty and finite for n ≥
2, we have coind
Per p ( Y, S ) = 0 for all primenumber p . (cid:3) See another example of above lemma in [TTY20, Lemma 4.1]
Proposition 4.5.
Let ( X, T ) be a dynamical system without fixed points.Then there exists a positive integer N and a positive number δ such that coind Per p ( X ( N, , δ ) , σ ) ≥ coind Per p ( X, T ) + 1 , for all prime numbers p .Proof. By Lemma 4.4, we pick a dynamical system (
Y, S ) without fixedpoints satisfying that coind
Per p ( Y, S ) = 0 for all prime number p . Thenby Corollary 3.3 (2), we havecoind Per p ( X ∗ Y, T ∗ S ) ≥ coind Per p ( X, T ) + coind
Per p ( Y, S ) + 1= coind
Per p ( X, T ) + 1 , (4 · p . On the other hand, by Lemma 4.2, there existsa dynamical system ( X ( N, , δ ) , σ ) and an equivariant continuous map f : ( X ∗ Y, T ∗ S ) → ( X ( N, , δ ) , σ ) . By Corollary 3.3 (1), we have(4 ·
2) coind
Per p ( X ∗ Y, T ∗ S ) ≤ coind Per p ( X ( N, , δ ) , σ ) , for all prime number p . Combing (4 ·
2) with (4 · Per p ( X ( N, , δ ) , σ ) ≥ coind Per p ( X, T ) + 1 , for all prime number p . (cid:3) Finite mean dimension
In this section, we show the existence of aperiodic dynamical systemswith finite mean dimension which do not have the marker property. Westate our result as follows.
Theorem 5.1.
There exists an aperiodic dynamical system with finitemean dimension which does not have the marker property.
Recall that the diameter of S is 1. Define a subsystem of ( S Z , σ ) by Z := { ( x n ) n ∈ Z ∈ S Z : ∀ n ∈ Z , either ρ ( x n − , x n ) ≥
12 or ρ ( x n , x n +1 ) ≥ } . Obviously, the dynamical system ( Z , σ ) has no fixed points and P p ( Z , σ ) = ∅ for all prime numbers p . By Proposition 4.5, there exists a positiveinteger N , an δ > p ,(5 ·
1) coind
Per p ( X ( N, , δ ) , σ ) ≥ coind Per p ( Z , σ ) + 1 . Fix such N and δ through the whole section.Recall that X ( N, k, δ ) = { ( x n ) n ∈ Z ∈ ( S N ) Z : ρ N ( x n , x n + k ) ≥ δ, ∀ n ∈ Z } . Define q ( m ) = m Y n =1 n to be the product of first m positive integers for m ≥
1. Obviously, wehave m · q ( m −
1) = q ( m ) for m ≥
2. Let X m := X ( N, q ( m ) , δ ) for m ≥ . Here and below, we denote by ( X m , T m ) := ( X m , σ ) for convenience.For m >
1, we define an equivariant continuous map θ m,m − : X m → ( S N ) Z by ( x k ) k ∈ Z m − X i =0 x k + i · q ( m − ! k ∈ Z . As we have mentioned before, S N = ( R / Z ) N is an abelian group.Since ρ N m − X i =0 x k + i · q ( m − , m − X i =0 x ( k + q ( m − i · q ( m − ! = ρ N m − X i =0 x k + i · q ( m − , m X i =1 x k + i · q ( m − ! = ρ N (cid:0) x k , x k + m · q ( m − (cid:1) = ρ N (cid:0) x k , x k + q ( m ) (cid:1) , ∀ k ∈ Z , INITE MEAN DIMESNION AND MARKER PROPERTY 11 we obtain that the image of X m under θ m,m − is contained in X m − .For m > n , we define θ m,n = θ m,m − ◦ θ m − ,m − ◦ . . . θ n +1 ,n to be anequivariant continuous map from X m to X n .Fix a = ( a k ) k ∈ Z ∈ ( S N ) Z . For m ≥
2, we define a map η m − ,m = η am − ,m : X m − → ( S N ) Z by η m − ,m (( x k ) k ∈ Z ) = ( y k ) k ∈ Z where y k = a k if 0 ≤ k ≤ ( m − · q ( m − − ,x k − ( m − q ( m − − P m − i =1 a k − i · q ( m − if ( m − · q ( m ) ≤ k ≤ q ( m ) − P n − i =0 (cid:0) x i · q ( m )+ q ( m − j − x i · q ( m )+ j (cid:1) + y j if k = n · q ( m ) + j with n > ≤ j ≤ q ( m ) − , P − i = n (cid:0) x i · q ( m )+ j − x i · q ( m )+ q ( m − j (cid:1) + y j if k = n · q ( m ) + j with n < ≤ j ≤ q ( m ) − . We show several properties of η m − ,m in the following lemmas. Lemma 5.2.
For m ≥ , η m − ,m ( X m − ) ⊂ X m .Proof. Let x ∈ X m − and y = η m − ,m ( x ). Let k = n · q ( m ) + j with n ∈ Z and 0 ≤ j ≤ q ( m ) −
1. We divide the proof in the followingthree cases according to the value of n .Case 1. n = 1. We have y k − y k − q ( m ) = x q ( m − j − x j + y j − y j = x q ( m − j − x j . Since x ∈ X m − , we have ρ N ( y k , y k − q ( m ) ) = ρ N ( x q ( m − j , x j ) ≥ δ. Case 2. n ≥
2. A simple computation shows that y k − y k − q ( m ) = n − X i =0 (cid:0) x i · q ( m )+ q ( m − j − x i · q ( m )+ j (cid:1) − n − X i =0 (cid:0) x i · q ( m )+ q ( m − j − x i · q ( m )+ j (cid:1) = x ( n − q ( m )+ q ( m − j − x ( n − · q ( m )+ j = x k − q ( m )+ q ( m − − x k − q ( m ) . It follows that ρ N ( y k , y k − q ( m ) ) = ρ N ( x k − q ( m )+ q ( m − , x k − q ( m ) ) ≥ δ. Case 3. n ≤
0. Similarly to Case 1 and Case 2, we have y k − y k − q ( m ) = x k − q ( m )+ q ( m − − x k − q ( m ) , and consequently ρ N ( y k , y k − q ( m ) ) ≥ δ. To sum up, we conclude that y ∈ X m and η m − ,m ( X m − ) ⊂ X m . (cid:3) By Lemma 5.2, we see that η m − ,m is the map from X m − to X m .Moreover, we show in the following that η m − ,m is indeed a right inversemap of θ m,m − . Lemma 5.3. θ m,m − ◦ η m − ,m = id , ∀ m ≥ . Proof.
Let x ∈ X m − and y = η m − ,m ( x ). Then we have θ m,m − ◦ η m − ,m ( x ) = θ m,m − ( y ) = m − X i =0 y k + i · q ( m − ! k ∈ Z . If 0 ≤ k ≤ q ( m − −
1, then m − X i =0 y k + i · q ( m − = m − X i =0 a k + i · q ( m − + x k − m − X i =1 a k +( m − − i ) q ( m − ! = x k . If k = s · q ( m ) + t · q ( m −
1) + j > q ( m −
1) for s >
0, 0 ≤ t ≤ m − ≤ j ≤ q ( m − −
1, then m − X i =0 y k + i · q ( m − = m − X i = t y s · q ( m )+ i · q ( m − j + t − X i =0 y ( s +1) q ( m )+ i · q ( m − j = m − X i = t s − X ℓ =0 (cid:0) x ℓ · q ( m )+( i +1) q ( m − j − x ℓ · q ( m )+ i · q ( m − j (cid:1) + t − X i =0 s X ℓ =0 (cid:0) x ℓ · q ( m )+( i +1) q ( m − j − x ℓ · q ( m )+ i · q ( m − j (cid:1) + m − X i =0 y i · q ( m − j = s − X ℓ =0 m − X i =0 (cid:0) x ℓ · q ( m )+( i +1) q ( m − j − x ℓ · q ( m )+ i · q ( m − j (cid:1) + t − X i =0 (cid:0) x s · q ( m )+( i +1) q ( m − j − x s · q ( m )+ i · q ( m − j (cid:1) + x j = s − X ℓ =0 (cid:0) x ( ℓ +1) · q ( m )+ j − x ℓ · q ( m )+ j (cid:1) + (cid:0) x s · q ( m )+ t · q ( m − j − x s · q ( m )+ j (cid:1) + x j = x s · q ( m )+ j − x j + x k − x s · q ( m )+ j + x j = x k . If k = s · q ( m ) + t · q ( m −
1) + j for s <
0, 0 ≤ t ≤ m − ≤ j ≤ q ( m − −
1, then by the similar computation of the casewhere s >
0, we have P m − i =0 y k + i · q ( m − = x k . This completes theproof. (cid:3) INITE MEAN DIMESNION AND MARKER PROPERTY 13
By Lemma 5.3, the map θ m,n are surjective maps from X m to X n for all m > n . Therefore, denote by lim ←− ( X n , T n ) the inverse limit of { ( X n , T n ) } n ∈ N via ( θ m,n ) n,m ∈ N ,m>n . Lemma 5.4.
The inverse limit lim ←− ( X n , T n ) is aperiodic.Proof. Suppose lim ←− ( X n , T n ) has a periodic point y of period n . Let π n :lim ←− ( X n , T n ) → ( X n , T n ) be the natural projection. Then x := π n ( y )has the period n in ( X n , T n ). It follows from n | q ( n ) that x = x q ( n ) .This is a contradiction to the definition of X n , that is, | x − x q ( n ) | > δ .Thus we conclude that lim ←− ( X n , T n ) is aperiodic. (cid:3) We will show later that lim ←− ( X n , T n ) is desired for Theorem 5.1.To this end, we need to do some preparation. Lindenstrauss [Lin99,Lemma 3.3] introduced a topological dynamics version of the Rokhlintower lemma in ergodic theory. Lemma 5.5.
Let ( X, T ) be a dynamical system having the markerproperty. For each positive number N there is a continuous function ϕ : X → R such that the set E := { x ∈ X : ϕ ( T x ) = ϕ ( x ) + 1 } satisfies that E ∩ T − n E = ∅ for all ≤ n ≤ N . As it is pointed out by Gutman [Gut15, Theorem 7.3], the existenceof a continuous function ϕ in Lemma 5.5 is indeed equivalent to themarker property.Define Y := { ( x n ) n ∈ Z ∈ S Z : ∀ n ∈ Z , either ρ ( x n − , x n ) = 1 or ρ ( x n , x n +1 ) = 1 } . Concerning with the system ( Y , σ ), Tsukamoto, Tsutaya and Yoshi-naga showed a necessary condition for the marker property [TTY20,Lemma 5.3]. We give the proof here for completeness. Lemma 5.6.
Let ( X, T ) be a dynamical system having marker prop-erty. Then there is an equivariant continuous map from ( X, T ) to ( Y , σ ) .Proof. By Lemma 5.5, there is a continuous function ϕ : X → R suchthat the set E := { x ∈ X : ϕ ( T x ) = ϕ ( x ) + 1 } satisfies that E ∩ T − E = ∅ . Let P : R → S be the natural projection.Set φ = P ◦ ϕ : X → S . Notice that(5 · x / ∈ E = ⇒ ρ ( φ ( T x ) , φ ( x )) = 1 . Define an equivariant continuous map f : X → S Z by x ( φ ( T n x )) n ∈ Z . Fix x ∈ X and n ∈ Z . Since E ∩ T − E = ∅ , we get that either T n x / ∈ E or T n +1 x / ∈ E . It follows from (5 ·
2) that either ρ ( φ ( T n +1 x ) , φ ( T n x )) = ρ ( φ ( T n +2 x ) , φ ( T n +1 x )) = 1. Since x and n are chosen arbitrarily,we obtain that the image of X under f is included in Y . This completesthe proof. (cid:3) Recall that η m − ,m is the map from X m − to X m for m ≥
2. Sincewe have d ( η m − ,m ( x ) , η m − ,m ( y )) ≤ q ( m ) − X k =( m − q ( m − k ρ N ( x k − ( m − q ( m − , y k − ( m − q ( m − )+ + ∞ X n =1 q ( m ) − X j =0 n · q ( m )+ j n − X i =0 ( ρ N ( x i · q ( m ) − q ( m − j , y i · q ( m ) − q ( m − j )+ ρ N ( x i · q ( m )+ j , y i · q ( m )+ j )) + −∞ X n = − q ( m ) − X j =0 | n · q ( m )+ j | · − X i = n ( ρ N ( x i · q ( m )+ j , y i · q ( m )+ j ) + ρ N ( x i · q ( m )+ q ( m − j , y i · q ( m )+ q ( m − j )) , the map η m − ,m is continuous for each m ≥
2. We define a continuousmap γ m : X m → lim ←− ( X n , T n ) by x ( θ m, ( x ) , θ m, ( x ) , . . . , θ m,m − ( x ) , x, η m,m +1 ( x ) , η m,m +2 ( x ) , . . . ) , where η m,m + n ( x ) = η m + n − ,m + n ◦ · · · η m +1 ,m +2 ◦ η m,m +1 ( x ). Lemma 5.7.
Let ( X n , T n ) be defined as above. Suppose there is anequivariant continuous map f : lim ←− ( X n , T n ) → ( Y , σ ) . Then thereexists an integer M and an equivariant continuous map g : ( X M , T M ) −→ ( Z , σ ) . Proof.
For convenience, we denote the inverse limit by ( X , T ) = lim ←− ( X n , T n ).Let π m : X → X m be the natural projection for m ∈ N . Let P : S Z →S be the projection on 0-th coordinate. Define φ = P ◦ f : X → S .Then f ( x ) = ( φ ( T n x )) n ∈ Z for any x ∈ X .Notice that there exists an integer M > · π M ( x ) = π M ( y ) = ⇒ ρ ( φ ( x ) , φ ( y )) < . Define a continuous map ϕ = φ ◦ γ M : X M → S where γ M : X M → X is defined as above. Now we define an equivariant continuous map g : X M → S Z by x ( ϕ ( T nM ( x ))) n ∈ Z . Since π M ◦ γ M = id and π M ◦ T = T M ◦ π M , it follows from (5 ·
3) that(5 · ρ ( φ ( γ M ( T nM ( x ))) , φ ( T n ( γ M ( x )))) < , ∀ n ∈ Z . INITE MEAN DIMESNION AND MARKER PROPERTY 15
Fix x ∈ X M and n ∈ Z . By definitions of Y and f , there exists an i ∈ { , } such that(5 · ρ ( φ ( T n + i ( γ M ( x ))) , φ ( T n + i +1 ( γ M ( x )))) = 1 . Combing (5 ·
5) with (5 · ρ (cid:0) φ ( γ M ( T n + iM x )) , φ ( γ M ( T n + i +1 M x )) (cid:1) ≥ ρ ( φ ( T n + i ( γ M ( x ))) , φ ( T n + i +1 ( γ M ( x )))) − ρ (cid:0) φ ( γ M ( T n + iM x )) , φ ( T n + i ( γ M ( x ))) (cid:1) − ρ (cid:0) φ ( γ M ( T n + i +1 M x )) , φ ( T n + i +1 ( γ M ( x ))) (cid:1) ≥ − −
14 = 12 . Since ϕ = φ ◦ γ M , we have that ρ (cid:0) ϕ ( T n + iM x ) , ϕ ( T n + i +1 M x ) (cid:1) ≥ . By definition of g and arbitrariness of n and x , we conclude that theimage of X M under g is contained in Z . This completes the proof. (cid:3) Now we present a result for general inverse limits.
Proposition 5.8.
Let { ( X n , T n ) } be a sequence of dynamical system.Let lim ←− ( X n , T n ) be the inverse limit of { ( X n , T n ) } n ∈ N via factor maps ( τ m,n ) m,n ∈ N ,m>n . Then mdim(lim ←− ( X n , T n )) ≤ sup n mdim( X n , T n ) . Proof.
Let π i : lim ←− ( X n , T n ) → X i be the projections. Pick a finite opencover A of lim ←− ( X n , T n ). Then there exists m = m ( A ) > A i of X i for 1 ≤ i ≤ m such that m _ i =1 π − i ( A i ) ≻ A . Notice that m _ i =1 π − i ( A i ) = m _ i =1 π − m ( τ − m,i ( A i )) = π − m m _ i =1 τ − m,i ( A i ) ! . Set B := W mi =1 τ − m,i ( A i ) which is a finite open cover of X m . Then wehave π − m ( B ) refines A . It follows that D ( ∨ Ni =0 T − i A ) N ≤ D ( ∨ Ni =0 T − i π − m ( B )) N = D ( π − m ( ∨ Ni =0 T − im ( B ))) N ≤ D ( ∨ Ni =0 T − im B ) N .
Therefore, it implies that mdim(lim ←− ( X n , T n )) ≤ sup n mdim( X n , T n ) . This completes the proof. (cid:3)
Now we prove the main result in this section.
Proof of Theorem 5.1.
Let ( X m , T m ) be the dynamical systems definedas above. Since X m ⊂ ( S N ) Z , for every m ∈ N , we havemdim( X m , T m ) ≤ mdim(( S N ) Z , σ ) ≤ N where the last inequality is due to [LW00, Proposition 3.1]. Let ( X , T ) :=lim ←− ( X n , T n ). By Lemma 5.4, the dynamical system ( X , T ) is aperiodic.By Proposition 5.8, we have mdim( X , T ) ≤ N . Suppose ( X , T ) hasthe marker property. By Lemma 5.6, there is an equivariant continu-ous map f : ( X , T ) → ( Y , σ ). Then by Lemma 5.7, there exists M > g : ( X M , T M ) → ( Z , σ ). It followsfrom Corollary 3.3 (1) thatcoind Per p ( X M , T M ) ≤ coind Per p ( Z , σ ) , for all prime numbers p > M . By Lemma 4.3, we getcoind Per p ( X , T ) ≤ coind Per p ( Z , σ ) , for all prime numbers p > M . This is a contradiction to the choice of N and δ , that is, (5 · X , T ) is an aperiodic dynamical systemhaving finite mean dimension which does not have the marker property.We complete the proof. (cid:3) Marker property and small mean dimension
Let (
X, T ) be a dynamical system. Let Z n := Z /n Z be a finiteabelian group for n ≥
1. We define the continuous map T n : X × Z n → X × Z n by T n ( x, k ) = ( ( x, k + 1) if 0 ≤ k ≤ n − , ( T x,
0) if k = n − . We call ( X × Z n , T n ) the n -time system over ( X, T ). Actually, thedynamical system ( X × { } , T nn ) is topologically conjugate to ( X, T ).It follows that mdim( X × Z n , T n ) = n mdim( X, T ).Now we investigate the relation between the marker property of adynamical system and the one of its n -time system. Proposition 6.1.
A dynamical system ( X, T ) has the marker propertyif and only if ( X × Z n , T n ) has the marker property.Proof. Suppose (
X, T ) has the marker property. Let N ≥
1. Let U ⊂ X be an open N -marker, i.e. U ∩ T k U = ∅ for 0 < k ≤ N and X = ∪ k ∈ Z T k U . It is easy to see that U ∩ T kn U = ∅ for 0 < k ≤ nN and X × Z n = ∪ k ∈ Z T kn U . Thus U is an open nN -marker of ( X × Z n , T n ).As N is chosen arbitrarily, we get that ( X × Z n , T n ) has the markerproperty.Now suppose ( X × Z n , T n ) has the marker property. Since ( X, T ) istopologically conjugate to ( X × { } , T nn ), it is sufficient to show that INITE MEAN DIMESNION AND MARKER PROPERTY 17 ( X × { } , T nn ) has the marker property. Fix N >
0. Let W be an open nN -marker of ( X × Z n , T n ). Let U := (cid:0) ∪ n − j =0 T jn ( W ) (cid:1) ∩ ( X × { } ) , which is an open set in X × { } . Notice that T knn U = (cid:0) ∪ n − j =0 T kn + jn ( W ) (cid:1) ∩ ( X × { } ) . By assumption, we have that T in ( W ) ∩ T jn ( W ) = ∅ whenever i = j and | i − j | ≤ nN . It follows that U ∩ T knn U = (cid:0) ∪ n − j =0 T jn ( W ) ∩ ∪ n − j =0 T kn + jn ( W ) (cid:1) ∩ ( X × { } ) = ∅ , for 0 < k ≤ N −
1. Moreover, ∪ k ∈ Z T knn U = (cid:0) ∪ j ∈ Z T jn ( W ) (cid:1) ∩ ( X ×{ } ) = ( X × Z n ) ∩ ( X ×{ } ) = X ×{ } . It means that U is a ( N − X ×{ } , T nn ). Since N is chosenarbitrarily, the system ( X × { } , T nn ) has the marker property. (cid:3) Now we present the main result in this section.
Theorem 6.2 (=Theorem 1.2) . For any η > , there exists an aperi-odic dynamical system with mean dimension small than η which doesnot have the marker property.Proof. By Theorem 1.2, there exists an aperiodic finite mean dimen-sional dynamical system ( X , T ) which does not have the marker prop-erty. Let n ≥ X × Z n , T n ) be the dynamical system definedas above. By Lemma 6.1, the dynamical system ( X × Z n , T n ) does nothave the marker property. Notice thatmdim( X × Z n , T n ) = 1 n mdim( X , T ) . We complete the proof by taking n arbitrarily large. (cid:3) Open problem
In this section, we discuss open problems related to the results of thecurrent paper. Firstly, we present a corollary of Theorem 6.2.
Corollary 7.1.
For any number η ∈ (0 , + ∞ ] , there exists an aperiodicdynamical system with mean dimension equal to η which does not havethe marker property.Proof. Let η ∈ (0 , + ∞ ]. By Theorem 6.2, there exists an aperiodicdynamical system ( X, T ) with mean dimension small than η whichdoes not have the marker property. Using the construction by Lin-denstrauss and Weiss [LW00, Page 10-11] (see also [LT14, Section3]), there is a minimal dynamical system ( Y, S ) with mean dimensionequal to η . By definition of mean dimension, it is easy to check thatmdim(( X, T ) ∪ ( Y, S )) = max { mdim( X, T ) , mdim( Y, S ) } = η . Since( X, T ) and (
Y, S ) are both aperiodic, the system (
X, T ) ∪ ( Y, S ) is also aperiodic. Moreover, (
X, T ) ∪ ( Y, S ) does not have the marker prop-erty. Indeed, if (
X, T ) ∪ ( Y, S ) has the marker property, then so doboth (
X, T ) and (
Y, S ). It is a contradiction because (
X, T ) does nothave the marker property. Thus the system (
X, T ) ∪ ( Y, S ) is what wedesire. (cid:3)
As it was discussed in [TTY20], one need certain additional conditionbesides the aperiodicity to guarantee the marker property. Due toCorollary 7.1, the condition of zero mean dimension seems plausible.Here we restate [TTY20, Conjecture 7.4] as follows.
Conjecture 7.2.
An aperiodic dynamical system with zero mean di-mension has the marker property.
Gutman [Gut15, Theorem 6.1] proved that every finite dimensionalaperiodic dynamical system has the marker property. Thus it reducesConjecture 7.2 to the case that aperiodic dynamical systems have zeromean dimension and are infinite dimensional. However, it seems diffi-cult and is widely open in such case.A dynamical system is said to have the small boundary property iffor any point one can find a neighborhood whose boundary is a nullset. Lindenstrauss [Lin99] proposed the following conjecture. As itis pointed out in [TTY20], Conjecture 7.2 is indeed equivalent to thefollowing conjecture proposed by Lindenstrauss.
Conjecture 7.3.
An aperiodic dynamical system with zero mean di-mension has the small boundary property.
Lindenstrauss [Lin99, Theorem 6.2] proved that if a dynamical sys-tem with zero mean dimension has the marker property then it has thesmall boundary property. Nevertheless, Conjecture 7.3 is still open ingeneral.
Acknowledgement
We thank Masaki Tsukamoto and Yonatan Gutman for valuable dis-cussion and remarks.
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