Symbolic dynamics in mean dimension theory
aa r X i v : . [ m a t h . D S ] O c t SYMBOLIC DYNAMICS IN MEAN DIMENSION THEORY
MAO SHINODA, MASAKI TSUKAMOTO
Abstract.
Furstenberg (1967) calculated the Hausdorff and Minkowski dimensions ofone-sided subshifts in terms of topological entropy. We generalize this to Z -subshifts.Our generalization involves mean dimension theory. We calculate the metric mean di-mension and mean Hausdorff dimension of Z -subshifts with respect to a subaction of Z . The resulting formula is quite analogous to Furstenberg’s theorem. We also calculatethe rate distortion dimension of Z -subshifts in terms of Kolmogorov–Sinai entropy. Introduction
Hausdorff and Minkowski dimensions of subshifts.
Let A be a finite set (al-phabet). We consider the one-sided infinite product A N = A × A × A × · · · with the shiftmap σ : A N → A N defined by σ (( x n ) n ∈ N ) = ( x n +1 ) n ∈ N . Take α >
1. We define a distance d on A N by d ( x, y ) = α − min { n | x n = y n } . Let
X ⊂ A N be a σ -invariant closed subset. Furstenberg [Fur67, Proposition III.1] calcu-lated the Hausdorff and Minkowski dimensions of X with respect to d :(1.1) dim H ( X , d ) = dim M ( X , d ) = h top ( X , σ )log α . Here h top ( X , σ ) is the topological entropy of ( X , σ ). The purpose of the paper is to extendthis result to higher rank actions .1.2. Mean dimension theory.
Mean dimension theory provides a meaningful frame-work for extending (1.1) to higher rank actions. This is the theory first introduced byGromov [Gro99] and further developed by Lindenstrauss–Weiss [LW00], Lindenstrauss[Lin99], and more recently Lindenstrauss and the second named author [LT19]. We re-view the basic ingredients here. (The precise definitions will be given in § Date : October 3, 2019.2010
Mathematics Subject Classification.
Key words and phrases. subshift, metric mean dimension, mean Hausdorff dimension, rate distortiondimension.M.S. was partially supported by Grant-in-Aid for JSPS Research Fellow, JSPS KAKENHI GrantNumber 17J03495. M.T. was partially supported by JSPS KAKENHI 18K03275.
A pair ( X , T ) is called a dynamical system if X is a compact metric space and T : X → X is a homeomorphism . Gromov [Gro99] defined mean topological dimension mdim( X , T ). This is a dynamical analogue of topological dimension, and it evaluatesthe number of parameters per iterate for describing the orbits of ( X , T ). As the namesuggested, the mean topological dimension is a topological invariant of dynamical systems.There are many important works around this quantity [LW00, Lin99, Gut15, GLT16, GT,LL18, Tsu18, MT19, LT19]. However mean topological dimension is not the right notionfor the purpose of this paper because Furstenberg’s theorem (1.1) concerns with Hausdorffand Minkowski dimensions, not topological one. (The topological dimension of a subshift X ⊂ A N is simply zero.)Let d be a metric (i.e. a distance function) on X . Lindenstrauss–Weiss [LW00] defined metric mean dimension mdim M ( X , T, d ). This is a dynamical analogue of Minkowskidimension. Lindenstrauss and the second named author [LT19] defined mean Haus-dorff dimension mdim H ( X , T, d ). This is a dynamical analogue of Hausdorff dimension.Metric mean dimension and mean Hausdorff dimension are metric dependent quantities.They provide a good framework for the purpose of the paper.It is well-known in geometric measure theory [Mat95] that metrical dimensions aredeeply connected to measure theory. In particular we can introduce the concept of (metricdependent) dimension for measure (see e.g. [R´en59, Youn82, KD94]). Similarly we canintroduce a mean dimensional quantity for invariant measures of dynamical systems. Let µ be a T -invariant Borel probability measure on X . Let X be a random variable takingvalues in X according to the law µ , and we consider the stochastic process { T n X } n ∈ Z .We denote by R ( d, µ, ε ) ( ε >
0) the rate distortion function of this stochastic process.This is the key quantity of Shannon’s rate distortion theory [Sh48, Sh59]. It evaluateshow many bits per iterate we need for describing the process within the distortion (withrespect to d ) bound by ε . Following Kawabata–Dembo [KD94], we define the upper andlower rate distortion dimensions by (1.2) rdim( X , T, d, µ ) = lim sup ε → R ( d, µ, ε )log(1 /ε ) , rdim( X , T, d, µ ) = lim inf ε → R ( d, µ, ε )log(1 /ε ) . When the upper and lower limits coincide, we denote the common value by rdim( X , T, d, µ ).Metric mean dimension, mean Hausdorff dimension and rate distortion dimension arerelated to each other. See Proposition 2.1 and Theorem 2.4 below.1.3. Statement of the main result.
Let A be a finite set as in § A Z index by Z . We define the shifts σ and σ on A Z by σ (( x m,n ) m,n ∈ Z ) = ( x m +1 ,n ) m,n ∈ Z , σ (( x m,n ) m,n ∈ Z ) = ( x m,n +1 ) m,n ∈ Z . We can also consider a non-invertible map T as in § T here forsimplicity. Throughout the paper, we assume that the base of the logarithm is two.
YMBOLIC DYNAMICS IN MEAN DIMENSION THEORY 3
Fix α > d on A Z by(1.3) d ( x, y ) = α − min {| u | ∞ | x u = y u } , where | u | ∞ = max( | m | , | n | ) for u = ( m, n ) ∈ Z . We call a closed subset X ⊂ A Z subshift if it is invariant under both σ and σ .The following is our main result. Theorem 1.1.
Let
X ⊂ A Z be a subshift. Then (1.4) mdim H ( X , σ , d ) = mdim M ( X , σ , d ) = 2 h top ( X , σ , σ )log α . Here h top ( X , σ , σ ) is the topological entropy of ( X , σ , σ ) . Moreover, if µ is a Borelprobability measure on X invariant under both σ and σ then rdim( X , σ , d, µ ) = 2 h µ ( X , σ , σ )log α . Here h µ ( X , σ , σ ) is the Kolmogorov–Sinai entropy of ( X , σ , σ ) with respect to the mea-sure µ . In particular if µ is a maximal entropy measure (i.e. h µ ( X , σ , σ ) = h top ( X , σ , σ ))then rdim( X , σ , d, µ ) coincides with the mean Hausdorff dimension and metric meandimension.The point of the statement is that we consider various mean dimensional quantitiesfor the action of σ , not the total Z -action generated by σ and σ . In other wordswe consider only σ and disregard σ . Nevertheless we can recover the entropy of thetotal Z -action. This might look a bit strange at first sight. But in fact it has thesame spirit with Furstenberg’s theorem (1.1). In (1.1), we consider the Hausdorff andMinkowski dimensions of one-sided subshifts. Hausdorff and Minkowski dimensions arepurely metric invariants and do not involve dynamics. So here we disregard the action atall. However we can recover the topological entropy. See Remark 1.2 (4) below for morebackgrounds behind the formulation of the theorem. Remark 1.2. (1) Subshifts
X ⊂ A Z are totally disconnected. So the mean topolog-ical dimension of ( X , σ ) is zero.(2) Probably some readers notice a slight difference between Furstenberg’s theorem(1.1) and our (1.4): Our formula involves the coefficient “2” wheres Furstenberg’stheorem does not. This difference comes from the point that Furstenberg’s theoremconsiders one-sided shifts (i.e. actions of N , not Z ). If we consider two-sided shifts,then we get a result completely analogous to (1.4).(3) Theorem 1.1 can be generalized to Z k -shifts and, probably some noncommutativegroup actions. But we stick to Z for simplicity of the exposition. MAO SHINODA, MASAKI TSUKAMOTO (4) A guiding principle behind our theorem is as follows: Let T : Z k ×X → X be a con-tinuous action of Z k on a compact metric space X . If T has some hyperbolicity-likeproperty, then we can control the mean dimensional quantities of the restrictionof T to subgroups G ⊂ Z k with rank G = k − k = 1, the subgroup G must be trivial. So, in particular, this principleclaims that we can control the dimensions of X if X admits an action of Z withsome “hyperbolicty”. Furstenberg’s theorem (1.1) is a typical example of suchresults because symbolic dynamics can be seen as an extreme case of hyperbolicdynamics. Theorem (1.1) corresponds to the case of k = 2 in this principle.Another manifestation of the above principle was given by the work of [MT19].They proved that if T : Z k × X → X is expansive then the mean topologicaldimension of T | G is finite for any rank ( k −
1) subgroups G ⊂ Z k . In particular,when k = 1, a compact metric space has finite topological dimension if it admitsan expansive action of Z . This is a classical theorem of Ma˜n´e [Ma79].(5) We consider the action of σ in Theorem 1.1. This corresponds to a study of theaction of the subgroup { ( n, | n ∈ Z } ⊂ Z . According to the principle in theabove (4), it is also natural to consider other rank-one subgroups. Namely weshould study various mean dimensional quantities for the action of σ a σ b for anynonzero ( a, b ) ∈ Z , which corresponds to the subgroup { ( an, bn ) | n ∈ Z } .Indeed we can calculate them. Take a nonzero ( a, b ) ∈ Z . Then for a subshift X ⊂ A Z we havemdim M ( X , σ a σ b , d ) = mdim H ( X , σ a σ b , d ) = 2( | a | + | b | ) h top ( X , σ , σ )log α , rdim( X , σ a σ b , d, µ ) = 2( | a | + | b | ) h µ ( X , σ , σ )log α . (1.5) Here d is the metric defined by (1.3) and µ is a Borel probability measure on X invariant under σ and σ .The factor 2( | a | + | b | ) in (1.5) has the following geometric meaning. For naturalnumbers M and N , we define Λ a,b ( M, N ) ⊂ Z as the set of points( an + x, bn + y ) , (0 ≤ n < N, | ( x, y ) | ∞ < M ) . Here n, x, y are integers. (Namely, we consider the parallel translations of ( − M, M ) along the segment { ( an, bn ) | ≤ n < N } . ) Then we have2( | a | + | b | ) = lim M →∞ (cid:18) lim N →∞ | Λ a,b ( M, N ) | M N (cid:19) ( | · | denotes the cardinality) . The square ( − M, M ) is the disk of radius M in the ℓ ∞ -norm | u | ∞ . The rele-vance of the ℓ ∞ -norm here comes from the point that the metric (1.3) uses it. Ifwe use a different metric, then we get a different result. For example, consider the YMBOLIC DYNAMICS IN MEAN DIMENSION THEORY 5 following metric ρ on A Z :(1.6) ρ ( x, y ) = α − min {√ m + n | x m,n = y m,n } . This metric uses the ℓ -norm √ m + n instead of the ℓ ∞ -norm. For this metricwe havemdim M ( X , σ a σ b , ρ ) = mdim H ( X , σ a σ b , ρ ) = 2 √ a + b · h top ( X , σ , σ )log α , rdim( X , σ a σ b , ρ, µ ) = 2 √ a + b · h µ ( X , σ , σ )log α . (1.7) The proofs of (1.5) and (1.7) are conceptually the same with the proof of Theorem1.1. However they become notationally more messy. So we decide to concentrateon the statement of Theorem 1.1. It clarifies the ideas in the simplest form. Acknowledgment.
The first proof we gave to Theorem 1.1 contained a gap. Elon Lin-denstrauss pointed out this, and he also kindly explained to us how to fix the gap. Wewould like to thank him for the help.2.
Preliminaries
The purpose of this section is to define the three dynamical dimensions (metric meandimension, mean Hausdorff dimension and rate distortion dimension) and explain someof their basic properties.2.1. Metric mean dimension and mean Hausdorff dimension.
Let ( X , d ) be acompact metric space. For ε > X , d, ε ) as the minimum natural number n such that X can be covered by open sets U , . . . , U n with diam U i < ε for all 1 ≤ i ≤ n .The upper and lower Minkowski dimensions of ( X , d ) are given bydim M ( X , d ) = lim sup ε → log X , d, ε )log(1 /ε ) , dim M ( X , d ) = lim inf ε → log X , d, ε )log(1 /ε ) . For s ≥ ε > H sε ( X , d ) = inf ( ∞ X i =1 (diam E i ) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X = ∞ [ n =1 E i with diam E i < ε for all i ≥ ) . Here we use the convention that 0 = 1 and diam( ∅ ) s = 0. Since X is compact, this isequal to the infimum of n X i =1 (diam U i ) s over all finite open covers { U , . . . , U n } of X with diam U i < ε for all 1 ≤ i ≤ n . We setdim H ( X , d, ε ) = sup { s ≥ | H sε ( X , d ) ≥ } . We do not use mean topological dimension in the paper. So we skip to define it.
MAO SHINODA, MASAKI TSUKAMOTO
The Hausdorff dimension of ( X , d ) is given bydim H ( X , d ) = lim ε → dim H ( X , d, ε ) . Given a homeomorphism T : X → X , we define metrics d TN ( N ≥
1) on X by d TN ( x, y ) = max ≤ n
This limit exists because log X , d TN , ε ) is suadditive in N . We define the upper andlower metric mean dimensions bymdim M ( X , T, d ) = lim sup ε → S ( X , T, d, ε )log(1 /ε ) , mdim M ( X , T, d ) = lim inf ε → S ( X , T, d, ε )log(1 /ε ) . When the upper and lower limits coincide, we denote the common value by mdim M ( X , T, d ).We define the upper and lower mean Hausdorff dimensions bymdim H ( X , T, d ) = lim ε → (cid:18) lim sup N →∞ dim H ( X , d N , ε ) N (cid:19) , mdim H ( X , T, d ) = lim ε → (cid:18) lim inf N →∞ dim H ( X , d N , ε ) N (cid:19) . When these two quantities are equal to each other, we denote the common value bymdim H ( X , T, d ).The following is the dynamical analogue of the fact that Minkowski dimension boundsHausdorff dimension. It was proved in [LT19, Proposition 3.2]. Proposition 2.1. mdim H ( X , T, d ) ≤ mdim H ( X , T, d ) ≤ mdim M ( X , T, d ) ≤ mdim M ( X , T, d ) . Remark 2.2.
Here is one remark about the notation. In the paper [LT19], the lowermean Hausdorff dimension played no role. So the upper mean Hausdorff dimension wassimply denoted by mdim H ( X , T, d ) in [LT19].2.2. Mutual information.
Let (Ω , P ) be a probability space. Let X and Y be mea-surable spaces, and let X : Ω → X and Y : Ω → Y be measurable maps. We want todefine their mutual information I ( X ; Y ) as the measure of the amount of information X and Y share. (This will be used in the definition of rate distortion function in the nextsubsection.) The basic reference is [CT06]. YMBOLIC DYNAMICS IN MEAN DIMENSION THEORY 7
Case 1: When X and Y are finite sets. In this case we set I ( X ; Y ) = H ( X ) + H ( Y ) − H ( X, Y ) = H ( X ) − H ( Y | X )= X x ∈X ,y ∈Y P ( X = x, Y = y ) log P ( X = x, Y = y ) P ( X = x ) P ( Y = y ) . (2.1)Here we have used the convention that 0 log(0 /a ) = 0 for all a ≥ Case 2: General case.
Let f : X → A and g : Y → B be measurable maps such that A and B are finite sets. Then we can define I ( f ◦ X ; g ◦ Y ) by (2.1). We define I ( X ; Y )as the supremum of I ( f ◦ X ; g ◦ Y ) over all finite range measurable maps f on X and g on Y . When X and Y are finite sets, this definition is compatible with (2.1). (Namelythe supremum is attained when f and g are the identity maps.)The mutual information is symmetric and nonnegative: I ( X ; Y ) = I ( Y ; X ) ≥
0. Thefollowing basic result immediately follows from the above definition.
Lemma 2.3 (Data-Processing inequality) . Let Z and W be measurable spaces. If f : X → Z and g : Y → W be measurable maps, then I ( f ( X ); g ( Y )) ≤ I ( X ; Y ) . Rate distortion theory.
Here we introduce rate distortion function. As Shannonentropy is the fundamental limit of lossless data compression, rate distortion function isthe fundamental limit of lossy data compression . A friendly introduction can be foundin [CT06, Chapter 10].Let ( X , T ) be a dynamical system with a metric d and an invariant Borel probabilitymeasure µ . We define the rate distortion function R ( d, µ, ε ) ( ε >
0) as the infimum of I ( X ; Y ) N , where N runs over natural numbers, X and Y = ( Y , . . . , Y N − ) are random variablesdefined on some probability space (Ω , P ) such that • X takes values in X according to the law µ . • Y , . . . , Y N − take values in X and satisfy(2.2) E N N − X n =0 d ( T n X, Y n ) ! < ε. The condition (2.2) means that Y = ( Y , . . . , Y N − ) approximates the stochastic process X, T X, . . . , T N − X within the averaged distortion bound by ε . We define the upper andlower rate distortion dimensions rdim( X , T, d, µ ) and rdim( X , T, d, µ ) by (1.2) in § R ( d, µ, ε ) is the minimum rate when we try to quantizethe process { T n X } n ∈ Z within the averaged distortion bound by ε . See [CT06, Chapter10], [Gra90, Chapter 11] and [ECG94, LDN79] for the precise meaning of this statement. We always assume that the σ -algebras of finite sets are the largest ones, i.e. the sets of all subsets. For example, expanding a given signal in a wavelet basis and discarding a small terms.
MAO SHINODA, MASAKI TSUKAMOTO
The rest of this subsection is not used in the proof of Theorem 1.1. We include thisfor providing readers a wider view of the subject. A metric d is said to have the tamegrowth of covering numbers if for any δ > ε → ε δ log X , d, ε ) = 0 . Note that this is purely a condition on the metric structure and does not involve dynamics.(2.3) is a mild condition. It is known ([LT19, Lemma 3.10]) that every compact metrizablespace admits a metric satisfying (2.3). For example, the metrics (1.3) and (1.6) on theshift space A Z satisfy (2.3). The following theorem [LT19, Proposition 3.2, Theorem 3.11]provides a link between rate distortion dimension and various mean dimensions. Here wedenote by M T ( X ) the set of all invariant Borel probability measures on X . Theorem 2.4. rdim( X , T, d, µ ) ≤ mdim M ( X , T, d ) , rdim( X , T, d, µ ) ≤ mdim M ( X , T, d ) . If d has the tame growth of covering numbers then mdim H ( X , T, d ) ≤ sup µ ∈ M T ( X ) rdim( X , T, d, µ ) . Proof of Theorem 1.1
First we recall the notations of § A Z is the Z -full shift on the alphabet (finite set) A with the shifts σ and σ . Fix α > d on A Z by d ( x, y ) = α − min {| u | ∞ | x u = y u } . Let
X ⊂ A Z be a subshift (closed shift-invariant set) with a Borel probability measure µ invariant under both σ and σ .The proof of Theorem 1.1 is divided into 4 steps:(1) Prove the upper bound on the upper metric mean dimensionmdim M ( X , σ , d ) ≤ h top ( X , σ , σ )log α . (2) Prove the lower bound on the lower mean Hausdorff dimensionmdim H ( X , σ , d ) ≥ h top ( X , σ , σ )log α . (3) Prove the upper bound on the upper rate distortion dimensionrdim( X , σ , d, µ ) ≤ h µ ( X , σ , σ )log α . (4) Prove the lower bound on the lower rate distortion dimensionrdim( X , σ , d, µ ) ≥ h µ ( X , σ , σ )log α . YMBOLIC DYNAMICS IN MEAN DIMENSION THEORY 9
Since we know mdim H ( X , σ , d ) ≤ mdim M ( X , σ , d ) by Proposition 2.1, the steps (1)and (2) show mdim H ( X , σ , d ) = mdim M ( X , σ , d ) = 2 h top ( X , σ , σ )log α . The steps (3) and (4) showrdim( X , σ , d, µ ) = 2 h µ ( X , σ , σ )log α . The steps (1) and (3) are easy. The step (2) is the most involved. The four steps areindependent of each other.For Ω ⊂ Z we denote by π Ω : X → A Ω the natural projection. As in § d σ N ( x, y ) = max ≤ n
0. In this section, intervals mean discrete intervals. Namely, for example, [ a, b ] = { a, a + 1 , . . . , b − , b } and ( a, b ) = { a + 1 , a +2 , . . . , b − } for integers a ≤ b .3.1. Step 1: Proof of mdim M ( X , σ , d ) ≤ h top ( X , σ , σ ) / log α . Let 0 < ε < M with α − M < ε ≤ α − M +1 . Then X , d σ N , ε ) ≤ | π ( − M,N + M ) × ( − M,M ) ( X ) | . (Here | · | denotes the cardinality.) Since ( M −
1) log α ≤ log(1 /ε ) < M log α ,mdim M ( X , σ , d ) = lim sup ε → (cid:18) lim N →∞ log X , d σ N , ε ) N log(1 /ε ) (cid:19) ≤ lim M →∞ (cid:18) lim N →∞ log | π ( − M,N + M ) × ( − M,M ) ( X ) | N ( M −
1) log α (cid:19) = 2 h top ( X , σ , σ )log α . Step 2: Proof of mdim H ( X , σ , d ) ≥ h top ( X , σ , σ ) / log α . First we prepare someterminologies about the geometry of Z . In this subsection rectangles mean sets of theform [ a, b ] × [ c, d ] in Z for integers a ≤ b and c ≤ d . For a rectangle R = [ a, b ] × [ c, d ] wedefine a new rectangle 3 R by3 R = [2 a − b, b − a ] × [2 c − d, d − c ] . We have | R | = (3 b − a + 1)(3 d − c + 1) ≤ | R | .For two rectangles R = [ a, b ] × [ c, d ] and R ′ = [ a ′ , b ′ ] × [ c ′ , d ′ ], we denote by R ≤ R ′ if b − a ≤ b ′ − a ′ and d − c ≤ d ′ − c ′ . This defines an order among rectangles. (Strictlyspeaking, this is a “pre-order” because R ≤ R ′ and R ′ ≤ R does not imply R = R ′ .)A set of rectangles { R , . . . , R n } is said to be totally ordered if any two elements arecomparable, i.e. for any R i and R j we have either R i ≤ R j or R j ≤ R i . The following trivial fact will be used later: Suppose { R , . . . , R n } is totally ordered. Ifa set of rectangles { R ′ , . . . , R ′ n ′ } has the property that each R ′ i is a parallel translation ofsome R j (namely R ′ i = u + R j for some u ∈ Z ) then { R ′ , . . . , R ′ n ′ } is also totally ordered.The next lemma is a kind of finite Vitali covering lemma ([EW11, Lemma 2.27]) adaptedto our situation. Lemma 3.1.
Suppose a set of rectangles { R , . . . , R n } is totally ordered. Then we canfind a disjoint subfamily { R i , . . . , R i m } satisfying R ∪ · · · ∪ R n ⊂ R i ∪ R i ∪ · · · ∪ R i m . Note that this implies | R i ∪ · · · ∪ R i m | ≥ | R ∪ · · · ∪ R n | . Proof.
We use a simple greedy algorithm. We first choose (one of) the largest rectangle,say R i . Next, suppose we have chosen R i , . . . , R i k . We choose as R i k +1 the largestrectangle disjoint to R i ∪ · · · ∪ R i k . If there is no such a rectangle, the algorithm stops.Suppose the algorithm stops after m steps. For any R j there exists R i k with R i k ≥ R j and R i k ∩ R j = ∅ . This implies R j ⊂ R i k . (cid:3) For two sets Ω , Λ ⊂ Z we define ∂ Λ Ω as the set of u ∈ Z such that u +Λ has non-emptyintersections both with Ω and Z \ Ω. We set Int Λ Ω = Ω \ ∂ Λ Ω. This is the set of u ∈ Ωwith u + Λ ⊂ Ω.Let R ⊂ Z be a rectangle. A subset C ⊂ X is called a cylinder over R if there is x ∈ X such that C is equal to the set of y ∈ X satisfying π R ( y ) = π R ( x ).Set s = 2 h top ( X , σ , σ )log α . Suppose mdim H ( X , σ , d ) < s . We would like to get a contradiction. We fix ε > H ( X , σ , d ) < s − ε . Lemma 3.2.
For any finite subset Λ ⊂ Z and any positive number L , we can findrectangles R , . . . , R M ⊂ Z and subsets C , . . . , C M ⊂ X such that • Each C m is a cylinder over R m and they satisfy X = S Mm =1 C m . • All the rectangles R m contain the origin, and they are all sufficiently large so that | ∂ Λ R m | < | R m | L , | R m | > L. • The rectangles R , . . . , R M are totally ordered and satisfy M X m =1 α − ( s − ε ) | R m | < . Proof.
We choose a natural number r such that YMBOLIC DYNAMICS IN MEAN DIMENSION THEORY 11 • Every r ≥ r satisfies ( s − ε ) r < ( s − ε )( r − • If a rectangle R = [ a, b ] × [ c, d ] ⊂ Z satisfies b − a ≥ r and d − c ≥ r then | ∂ Λ R | < | R | L , | R | > L. From mdim H ( X , σ , d ) < s − ε , we can find N > N dim H ( X , d σ N , α − r ) < s − ε. This implies that there exists a covering X = E ∪ · · · ∪ E M satisfyingdiam( E m , d σ N ) < α − r ( ∀ ≤ m ≤ M ) , M X m =1 (diam( E m , d σ N )) ( s − ε ) N < . Set α − r m := diam( E m , d σ N ). Then r m is a natural number with r m > r . Choose a point x m from each E m , and let C m ⊂ X be a cylinder over the rectangle R m := [ − r m + 1 , N + r m − × [ − r m + 1 , r m − C m = π − R m ( π R m ( x m )). Then E m ⊂ C m and hence X = C ∪ · · · ∪ C m . Therectangles R m are totally ordered ( R m ≤ R m ′ if and only if r m ≤ r m ′ ).Recall that r m > r for all 1 ≤ m ≤ M . From the choice of r , | ∂ Λ R m | < | R m | L , | R m | > L. From | R m | = ( N + 2 r m − r m − ≥ N (2 r m − s − ε ) | R m | ≥
12 ( s − ε ) N (2 r m − >
12 ( s − ε ) N (2 r m ) by the choice of r = ( s − ε ) N r m . Hence α − ( s − ε ) | R m | < α − ( s − ε ) Nr m = (diam( E m , d σ N )) ( s − ε ) N . Therefore M X m =1 α − ( s − ε ) | R m | < M X m =1 (diam( E m , d σ N )) ( s − ε ) N < . (cid:3) We choose a real number 0 < δ < / p satisfying the followingconditions.(3.1) (cid:18) (cid:19) p < δ, H ( δ ) + δ log p < ε α, | A | δ < α ε/ . Here H ( δ ) = − δ log δ − (1 − δ ) log(1 − δ ). (Recall that the base of the logarithm is two.)The first condition is satisfied for p ≈ log(1 /δ ). Then we choose a sufficiently small δ satisfying the second and third conditions. By using Lemma 3.2 iteratively, we find rectangles R i,m and subsets C i,m ⊂ X for i = 1 , . . . , p and m = 1 , . . . , M i (where M i is a natural number depending on i ) satisfyingthe following conditions.(a) Each C i,m is a cylinder over R i,m . For each 1 ≤ i ≤ p we have X = S M i m =1 C i,m .(b) For each 1 ≤ i ≤ p , the rectangles R i, , R i, , . . . , R i,M i are totally ordered andsatisfy(3.2) M i X m =1 α − ( s − ε ) | R i,m | < . (c) All the rectangles R i,m contain the origin and they satisfy | R i,m | > /δ .(d) Set ˆ R i = S M i m =1 R i,m . Then for all j < i and m = 1 , . . . , M j we have | ∂ ˆ R i R j,m | < δ | R j,m | . Roughly speaking, the condition (d) means that the rectangles in one level (say, j ) aremuch larger than the rectangles in higher levels (say, i > j ). The construction goes fromthe level p to the bottom. First, by Lemma 3.2, we construct R p,m and C p,m . Next, byusing the lemma again, we construct R p − ,m and C p − ,m . We continue this process untilwe come to the first level ( R ,m and C ,m ). The condition (d) connects the constructionsin different levels. Lemma 3.3. If N > is sufficiently large then the following statement holds. For each x ∈ X we can choose a subset D ( x ) ⊂ { ( u, i, m ) | u ∈ [0 , N − , ≤ i ≤ p, ≤ m ≤ M i } such that (1) For ( u, i, m ) ∈ D ( x ) , we have σ u ( x ) ∈ C i,m and u + R i,m ⊂ [0 , N − . (2) If ( u, i, m ) and ( u ′ , i ′ , m ′ ) are two different elements of D ( x ) , then ( u + R i,m ) ∩ ( u ′ + R i ′ ,m ′ ) = ∅ . In particular (recall that R i,m contain the origin), u = u ′ . (3) We have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [0 , N − \ [ ( u,i,m ) ∈ D ( x ) ( u + R i,m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δN . Proof.
Let N be sufficiently large so that(3.3) (cid:12)(cid:12) ∂ ˆ R i [0 , N − (cid:12)(cid:12) < δ N for all 1 ≤ i ≤ p. Here recall that ˆ R i = S M i m =1 R i,m . Fix x ∈ X . Set E = [0 , N − . We will inductivelyconstruct E ⊃ E ⊃ E ⊃ · · · ⊃ E p .Suppose we have defined E , E , . . . , E i − . Consider the following set of rectangles:(3.4) { u + R i,m | u ∈ E i − , m ∈ [1 , M i ] with σ u ( x ) ∈ C i,m and u + R i,m ⊂ E i − } . YMBOLIC DYNAMICS IN MEAN DIMENSION THEORY 13
Since R i, , . . . , R i,M i are totally ordered, so is (3.4). (Here the point is that i is fixed.)The rectangles (3.4) cover Int ˆ R i E i − . Then by Lemma 3.1, we can find a subset D i ( x ) ⊂ { ( u, m ) | u ∈ E i − , ≤ m ≤ M i } such that • For ( u, m ) ∈ D i ( x ), we have σ u ( x ) ∈ C i,m and u + R i,m ⊂ E i − . • If ( u, m ) and ( u ′ , m ′ ) are two different elements of D i ( x ) then ( u + R i,m ) ∩ ( u ′ + R i,m ′ ) = ∅ . • The rectangles u + R i,m , ( u, m ) ∈ D i ( x ), cover at least one-ninth of Int ˆ R i E i − :(3.5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ ( u,m ) ∈ D i ( x ) ( u + R i,m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12) Int ˆ R i E i − (cid:12)(cid:12) . We set E i = E i − \ [ ( u,m ) ∈ D i ( x ) ( u + R i,m ) . We define D ( x ) by D ( x ) = { ( u, i, m ) | ≤ i ≤ p, ( u, m ) ∈ D i ( x ) } . The properties (1) and (2) of D ( x ) immediately follow from the construction. The prop-erty (3) is equivalent to the claim that | E p | < δN . We will prove this.Suppose | E p | ≥ δN . Then we also have | E i − | ≥ δN for all 1 ≤ i ≤ p . We estimate (cid:12)(cid:12) ∂ ˆ R i E i − (cid:12)(cid:12) for 1 ≤ i ≤ p . We have ∂ ˆ R i E i − ⊂ ∂ ˆ R i [0 , N − ∪ i − [ j =1 [ ( u,m ) ∈ D j ( x ) ∂ ˆ R i ( u + R j,m ) . Recall (3.3) and | ∂ ˆ R i R j,m | < ( δ/ | R j,m | for j < i by the condition (d) of the choice of R i,m . Then (cid:12)(cid:12) ∂ ˆ R i E i − (cid:12)(cid:12) ≤ (cid:12)(cid:12) ∂ ˆ R i [0 , N − (cid:12)(cid:12) + i − X j =1 X ( u,m ) ∈ D j ( x ) (cid:12)(cid:12) ∂ ˆ R i ( u + R j,m ) (cid:12)(cid:12) < δ N + δ i − X j =1 X ( u,m ) ∈ D j ( x ) | u + R j,m | . The rectangles u + R j,m , 1 ≤ j ≤ i − u, m ) ∈ D j ( x ), are disjoint and contained in[0 , N − . Therefore i − X j =1 X ( u,m ) ∈ D j ( x ) | u + R j,m | ≤ N . Thus (cid:12)(cid:12) ∂ ˆ R i E i − (cid:12)(cid:12) < ( δ/ N . Since we assumed | E i − | ≥ δN , we have (cid:12)(cid:12) ∂ ˆ R i E i − (cid:12)(cid:12) < (1 / | E i − | . Namely (cid:12)(cid:12) Int ˆ R i E i − (cid:12)(cid:12) > | E i − | . From (3.5), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ ( u,m ) ∈ D i ( x ) ( u + R i,m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12) Int ˆ R i E i − (cid:12)(cid:12) > | E i − | . So we get | E i | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E i − \ [ ( u,m ) ∈ D i ( x ) ( u + R i,m ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < | E i − | . This holds for all 1 ≤ i ≤ p . Therefore | E p | < (cid:18) (cid:19) p | E | = (cid:18) (cid:19) p N . Recall that p satisfies (17 / p < δ by (3.1). So | E p | < δN . This is a contradiction. (cid:3) In the rest of this subsection, N is assumed to be so large that the statement of Lemma3.3 holds. For each x ∈ X we define D ( x ) ⊂ [0 , N − × [1 , p ] as the set of ( u, i ) ∈ [0 , N − × [1 , p ] such that there exists m ∈ [1 , M i ] with ( u, i, m ) ∈ D ( x ). (Notice thatthe sets D ( x ) and D ( x ) depend on N . So it might be better to use the notations D ( N ) ( x )and D ( N ) ( x ). But we prefer the simpler ones here.) Lemma 3.4. If N is sufficiently large then the number of possibilities of D ( x ) is boundedas follows: |{ D ( x ) | x ∈ X }| < α ( ε/ N . Proof.
We use the well-known bound on the binomial coefficient:(3.6) (cid:18) nk (cid:19) ≤ nH ( k/n ) . This follows from1 = (cid:26) kn + (cid:18) − kn (cid:19)(cid:27) n ≥ (cid:18) nk (cid:19) (cid:18) kn (cid:19) k (cid:18) − kn (cid:19) n − k = (cid:18) nk (cid:19) − nH ( k/n ) . Let x ∈ X and set D ( x ) = { ( u , i , m ) , . . . , ( u k , i k , m k ) } . (Then we have D ( x ) = { ( u , i ) , . . . , ( u k , i k ) } .) By (1) and (2) of Lemma 3.3, u , . . . , u k are different from eachother, and the rectangles u + R i ,m , . . . , u k + R i k ,m k are disjoint and contained in [0 , N − . Since | R i,m | > /δ (the condition (c) of the choice of R i,m ), we have k < δN . YMBOLIC DYNAMICS IN MEAN DIMENSION THEORY 15
Then the number of possibilities of D ( x ) is bounded by (cid:26)(cid:18) N (cid:19) + (cid:18) N (cid:19) + · · · + (cid:18) N ⌊ δN ⌋ (cid:19)(cid:27)| {z } choices of u , . . . , u k × p δN |{z} choices of i , . . . , i k ≤ N · N H ( δ ) × p δN by (3.6)= N · N ( H ( δ )+ δ log p ) . We assumed H ( δ ) + δ log p < ( ε/
8) log α in (3.1). Hence, if N is sufficiently large then N · N ( H ( δ )+ δ log p ) < N ( ε/
8) log α = α ( ε/ N . (cid:3) Take a subset E ⊂ [0 , N − × [1 , p ] such that there exists x ∈ X with D ( x ) = E . Wedenote by X E the set of x ∈ X with D ( x ) = E . Let E = { ( u , i ) , ( u , i ) , . . . , ( u k , i k ) } . Lemma 3.5. (cid:12)(cid:12) π [0 ,N − ( X E ) (cid:12)(cid:12) · α − ( s − ε ) N ≤ | A | δN M i X m =1 α − ( s − ε ) | R i ,m | × · · · × M ik X m =1 α − ( s − ε ) | R ik,m | . (3.7) Proof.
For m = ( m , . . . , m k ) ∈ [1 , M i ] ×· · ·× [1 , M i k ], we denote by X E, m ⊂ X E the set of x ∈ X E with D ( x ) = { ( u , i , m ) , ( u , i , m ) , . . . , ( u k , i k , m k ) } . We have σ u j ( x ) ∈ C i j ,m j for x ∈ X E, m . Hence, over each rectangle u j + R i j ,m j , the value of π u j + R ij,mj ( x ) ( x ∈ X E, m )is fixed. (Namely we have π u j + R ij,mj ( x ) = π u j + R ij,mj ( x ′ ) for any two x, x ′ ∈ X E, m .)Therefore we have | π [0 ,N − ( X E, m ) | ≤ | A || [0 ,N − \ S kj =1 ( u j + R ij,mj ) | < | A | δN . Here the second inequality follows from the condition (3) of Lemma 3.3. We decomposethe left-hand side of (3.7) as (cid:12)(cid:12) π [0 ,N − ( X E ) (cid:12)(cid:12) · α − ( s − ε ) N = X m (cid:12)(cid:12) π [0 ,N − ( X E, m ) (cid:12)(cid:12) · α − ( s − ε ) N ≤ X m with X E, m = ∅ | A | δN · α − ( s − ε ) N . (3.8)Take m = ( m , . . . , m k ) ∈ [1 , M i ] ×· · ·× [1 , M i k ] with X E, m = ∅ . The rectangles u j + R i j ,m j (1 ≤ j ≤ k ) are disjoint and contained in [0 , N − by the conditions (1) and (2) ofLemma 3.3. Hence N ≥ k X j =1 | R i j ,m k | . So α − ( s − ε ) N ≤ k Y j =1 α − ( s − ε ) | R ij,mj | . Plugging this into (3.8), we get (cid:12)(cid:12) π [0 ,N − ( X E ) (cid:12)(cid:12) · α − ( s − ε ) N ≤ X m | A | δN k Y j =1 α − ( s − ε ) | R ij,mj | . The right-hand side is equal to | A | δN M i X m =1 α − ( s − ε ) | R i ,m | × · · · × M ik X m =1 α − ( s − ε ) | R ik,m | . (cid:3) We continue the estimates: (cid:12)(cid:12) π [0 ,N − ( X E ) (cid:12)(cid:12) · α − ( s − ε ) N ≤ | A | δN M i X m =1 α − ( s − ε ) | R i ,m | × · · · × M ik X m =1 α − ( s − ε ) | R ik,m | < | A | δN by (3.2) < α ( ε/ N since we assumed | A | δ < α ε/ in (3.1) . The number of choices of E ⊂ [0 , N − × [1 , p ] with X E = ∅ is bounded by α ( ε/ N if N is sufficiently large (Lemma 3.4). Then (cid:12)(cid:12) π [0 ,N − ( X ) (cid:12)(cid:12) · α − ( s − ε ) N = X E with X E = ∅ (cid:12)(cid:12) π [0 ,N − ( X E ) (cid:12)(cid:12) · α − ( s − ε ) N < α ( ε/ N × α ( ε/ N = α ( ε/ N . Therefore (cid:12)(cid:12) π [0 ,N − ( X ) (cid:12)(cid:12) < α ( s − ε ) N . Namely log | π [0 ,N − ( X ) | N < (cid:16) s − ε (cid:17) log α. Letting N → ∞ h top ( X , σ , σ ) ≤ (cid:16) s − ε (cid:17) log α < s log α = h top ( X , σ , σ ) . This is a contradiction.
Remark 3.6. (1) The above proof (in particular, see the proof of Lemma 3.2) alsoshows a (seemingly) slightly stronger statement thatlim ε → (cid:18) inf N ≥ dim H ( X , d σ N , ε ) N (cid:19) ≥ h top ( X , σ , σ )log α . YMBOLIC DYNAMICS IN MEAN DIMENSION THEORY 17
Combined with Step 1, the both sides actually coincide. However we do not knowwhether the left-hand side is an important quantity or not.(2) The above proof (in particular, the use of covering argument) is motivated bythe proof of the Shannon–McMillan–Breiman theorem (see, e.g. [OW83, Rud90,Lin01]). We expect that there is a proof more directly using the Shannon–McMillan–Breiman theorem (or related measure theoretic ideas) although we havenot found it so far.3.3.
Step 3: Proof of rdim( X , σ , d, µ ) ≤ h µ ( X , σ , σ ) / log α . Let X be a randomvariable taking values in X and obeying µ . Let 0 < ε < M > α − M < ε ≤ α − M +1 as in Step 1. Let N >
0. For each point x ∈ π ( − M,N + M ) × ( − M,M ) ( X ) wechoose q ( x ) ∈ X with π ( − M,N + M ) × ( − M,M ) ( q ( x )) = x . Set X ′ = q (cid:0) π ( − M,N + M ) × ( − M,M ) ( X ) (cid:1) and Y = ( X ′ , σ X ′ , σ X ′ , . . . , σ N − X ′ ). Then1 N N − X n =0 d ( σ n X, Y n ) = 1 N N − X n =0 d ( σ n X, σ n X ′ ) ≤ α − M < ε.I ( X ; Y ) ≤ H ( Y ) = H ( X ′ ) = H (cid:8) ( X u ) u ∈ ( − M,N + M ) × ( − M,M ) (cid:9) . So R ( d, µ, ε ) ≤ I ( X ; Y ) N ≤ N H (cid:8) ( X u ) u ∈ ( − M,N + M ) × ( − M,M ) (cid:9) ,R ( d, µ, ε )log(1 /ε ) ≤ M log(1 /ε ) · N M H (cid:8) ( X u ) u ∈ ( − M,N + M ) × ( − M,M ) (cid:9) . We first take the limit with respect to N and next the limit with respect to ε . Noting M/ log(1 /ε ) → / log α , we getrdim( X , σ , d, µ ) ≤ h µ ( X , σ , σ )log α . Step 4: Proof of rdim( X , σ , d, µ ) ≥ h µ ( X , σ , σ ) / log α . We need the followinglemma.
Lemma 3.7.
Let N ≥ and B a finite set. Let X = ( X , . . . , X N − ) and Y =( Y , . . . , Y N − ) be random variables taking values in B N (namely, each X n and Y n takesvalues in B ) such that for some < δ < / E ( the number of ≤ n < N with X n = Y n ) < δN. Then I ( X ; Y ) > H ( X ) − N H ( δ ) − δN log | B | , where H ( δ ) = − δ log δ − (1 − δ ) log(1 − δ ) as in Step 2. Proof.
The proof is close to [LT18, Lemma 17]. Let Z n = 1 { X n = Y n } and Z = { ≤ n
0. Let Y = ( Y , . . . , Y N − ) be a random variable taking valuesin X N and satisfying E N N − X n =0 d ( σ n X, Y n ) ! < ε. We estimate I ( X ; Y ) from below. Take M ≥ δα − M − < ε ≤ δα − M . For0 ≤ n < N , we set X ′ n = π { n }× [ − M,M ] ( X ) = ( X n,m ) − M ≤ m ≤ M , Y ′ n = π { }× [ − M,M ] ( Y n ) = (( Y n ) ,m ) − M ≤ m ≤ M . YMBOLIC DYNAMICS IN MEAN DIMENSION THEORY 19 If X ′ n = Y ′ n for some n then d ( σ n X, Y n ) ≥ α − M . So E d ( σ n X, Y n ) ≥ α − M P ( X ′ n = Y ′ n ) andhence E (the number of 0 ≤ n < N with X ′ n = Y ′ n ) = N − X n =0 P ( X ′ n = Y ′ n ) ≤ α M E N − X n =0 d ( σ n X, Y n ) ! < α M εN ≤ δN. Apply Lemma 3.7 to X ′ n and Y ′ n with B = A M +1 : I ( X ′ , . . . , X ′ N − ; Y ′ , . . . , Y ′ N − ) >H ( X ′ , . . . , X ′ N − ) − N H ( δ ) − δN (2 M + 1) log | A | . By the data-processing inequality (Lemma 2.3), I ( X ; Y ) ≥ I ( X ′ , . . . , X ′ N − ; Y ′ , . . . , Y ′ N − ) . Therefore I ( X ; Y ) N ≥ H (cid:8) ( X u ) u ∈ [0 ,N ) × [ − M,M ] (cid:9) N − H ( δ ) − δ (2 M + 1) log | A | . This holds for any
N >
0. So R ( d, µ, ε ) ≥ inf N> H (cid:8) ( X u ) u ∈ [0 ,N ) × [ − M,M ] (cid:9) N − H ( δ ) − δ (2 M + 1) log | A | = lim N →∞ H (cid:8) ( X u ) u ∈ [0 ,N ) × [ − M,M ] (cid:9) N − H ( δ ) − δ (2 M + 1) log | A | . We divide this by log(1 /ε ) and take the limit ε →
0. Noting log(1 /ε ) < log(1 /δ ) + ( M +1) log α (here δ has been fixed), we getrdim( X , σ , d, µ ) ≥ h µ ( X , σ , σ )log α − δ log | A | log α . Here we have used h µ ( X , σ , σ ) = lim N,M →∞ H (cid:8) ( X u ) u ∈ [0 ,N ) × [ − M,M ] (cid:9) N (2 M + 1) . Take the limit δ →
0. We get rdim( X , σ , d, µ ) ≥ h µ ( X , σ , σ ) / log α . References [CT06] T. M. Cover, J. A. Thomas, Elements of information theory, second edition, Wiley, New York,2006.[ECG94] M. Effros, P. A. Chou, G. M. Gray, Variable-rate source coding theorems for stationary noner-godic sources, IEEE Trans. Inf. Theory vol. 40, pp. 1920-1925, 1994.[EW11] M. Einsiedler, T. Ward, Ergodic theory with a view towards number theory, Graduate Texts inMathematics , Springer, London. [Fur67] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantineapproximation, Math. Systems Theory (1967) 1-49.[Gra90] R.M. Gray, Entropy and information theory, New York, Springer-Verlag, 1990.[Gro99] M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps: I,Math. Phys. Anal. Geom. vol. 2 pp. 323-415, 1999.[Gut15] Y. Gutman, Mean dimension and Jaworski-type theorems, Proceedings of the London Mathe-matical Society (2015) 831-850.[GLT16] Y. Gutman, E. Lindenstrauss, M. Tsukamoto, Mean dimension of Z k -actions, Geom. Funct.Anal. Issue 3 (2016) 778-817.[GQT] Y. Gutman, Y. Qiao, M. Tsukamoto, Application of signal analysis to the embedding problem of Z k -actions, arXiv:1709.00125, to appear in Geom. Funct. Anal.[GT] Y. Gutman , M. Tsukamoto, Embedding minimal dynamical systems into Hilbert cubes, preprint,arXiv:1511.01802.[KD94] T. Kawabata and A. Dembo, The rate distortion dimension of sets and measures, IEEE Trans.Inf. Theory, vol. 40, no. 5, pp. 1564-1572, Sep. 1994.[LDN79] A. Leon-Garcia, L. D. Davisson, D. L. Neuhoff, New results on coding of stationary nonergodicsources, IEEE Trans. Inform. Theory, vol. 25, pp. 137-144, 1979.[LL18] H. Li, B. Liang, Mean dimension, mean rank and von Neumann–L¨uck rank, J. Reine Angew.Math. (2018) 207-240.[Lin99] E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes´Etudes Sci. Publ. Math. vol. 89 pp. 227-262, 1999.[Lin01] E. Lindenstrauss, Pointwise theorems for amenable groups, Invent. math. (2001) 259-296.[LT18] E. Lindenstrauss, M. Tsukamoto, From rate distortion theory to metric mean dimension: varia-tional principle, IEEE Trans. Inf. Theory, vol. 64, No. 5, pp. 3590-3609, May, 2018.[LT19] E. Lindenstrauss, M. Tsukamoto, Double variational principle for mean dimension, Geom. Funct.Anal. (2019) 1048-1109.[LW00] E. Lindenstrauss, B. Weiss, Mean topological dimension, Israel J. Math. vol. 115 pp. 1-24, 2000.[Ma79] R. Ma˜n´e, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. (1979) 313-319.[Mat95] P. Mattila, Geometry of sets and measures in Euclidean spaces, Fractals and rectifiability, Cam-bridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995.[MT19] T. Meyerovitch, M. Tsukamoto, Expansive multiparameter actions and mean dimension, Trans.Amer. Math. Soc. (2019), 7275-7299[OW83] D. S. Ornstein, B. Weiss, The Shannon–McMillan–Breiman theorem for a class of amenablegroups, Israel J. Math. (1983) 53-60.[R´en59] A. R´enyi, On the dimension and entropy of probability distributions, Acta Math. Sci. Hung. vol.10, pp. 193-215, 1959.[Rud90] D. J. Rudolph, Fundamentals of measurable dynamics, Clarendon Press, Oxford, 1990.[Sh48] C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. (1948) 379-423,623-656.[Sh59] C. E. Shannon, Coding theorems for a discrete source with a fidelity criterion, IRE Nat. Conv.Rec., Pt. 4, pp. 142-163, 1959.[Tsu18] M. Tsukamoto, Mean dimension of the dynamical system of Brody curves, Invent. math. (2018) 935-968. YMBOLIC DYNAMICS IN MEAN DIMENSION THEORY 21 [Youn82] L.-S. Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems (1982) 109-124. Mao ShinodaDepartment of Human Coexistence, Graduate School of Human and EnvironmentalStudies, Kyoto University, Yoshida-Nihonmaths-cho, Sakyo-ku, Kyoto, 606-8501, Japan
Email : [email protected] Masaki TsukamotoDepartment of Mathematics, Kyushu University, Moto-oka 744, Nishi-ku, Fukuoka 819-0395, Japan
E-mail ::