Rate distortion theory, metric mean dimension and measure theoretic ...

arXiv:1707.05762v1 [math.DS] 18 Jul 2017

RATE DISTORTION THEORY, METRIC MEAN DIMENSION AND MEASURE THEORETIC ENTROPY ANIBAL VELOZO AND RENATO VELOZO Abstract. We prove a variational principle for the metric mean dimension analog to the one in [LT]. Instead of using the rate distortion function we use the function hµ pǫ, T, δq that is closely related to the entropy hµ pT q of µ. Our formulation has the advantage of being, in the authors opinion, more natural when doing computations. As a corollary we obtain a proof of the standard variational principle. We also obtain some relations between the rate distortion function with our function r hµ pǫ, T, δq, a modification of hµ pǫ, T, δq when replacing the dynamical metrics with the average dynamical metrics. Using our methods we also reprove the main result in [LT]. We will explain how to construct homeomorphisms on closed manifolds with maximal metric mean dimension. We end this paper with some questions that naturally arise from this work.

1. Introduction The topological entropy is a fundamental quantity that allows us to quantify the chaoticity of a dynamical system. If the ambient space is compact and the dynamics is Lipschitz, then the topological entropy is finite. On the other hand, if the dynamics is just continuous, the topological entropy might be infinite. In fact, K. Yano proved in [Yan] that on a closed manifold of dimension at least two the topological entropy is infinite for generic homeomorphisms. It is then natural to consider a dynamical quantity that distinguishes systems with infinite topological entropy. The mean dimension is a meaningful quantity when the topological entropy is infinite. This invariant was introduced by Gromov in [Gro], and further studied by E. Lindenstrauss and B. Weiss in [LW]. This invariant has found many applications to embedding problems, in other words to the problem of when a dynamical system can be embedded into another or not, see for instance [L],[GLT] and references therein. In this paper we will mainly focus on the metric mean dimension. This is an invariant of the dynamical system defined in [LW], which in contrast with the topological entropy might depend on the metric on the ambient space. For completeness we proceed to define the relevant quantities. Let pX , dq be a compact metric space and T : X Ñ X a continuous map. Define Nd pn, ǫq as the maximal cardinality of a pn, ǫq-separated subset of X and SpX , d, ǫq “ lim sup nÑ8

1 log Nd pn, ǫq. n

For precise definitions see Section 2. The topological entropy is defined as htop pX , T q “ lim SpX , d, ǫq. ǫÑ0

Date: July 19, 2017. 1

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A. VELOZO AND R. VELOZO

The topological entropy is known to be independient of the compatible metric d since X is compact. The upper metric mean dimension is defined as mdimpX , d, T q “ lim sup ǫÑ0

SpX , d, ǫq . | log ǫ|

Recently E. Lindenstrauss and M. Tsukamoto in [LT] established a variational principle for the metric mean dimension. In this formula the role of the measure theoretic entropy is replaced by the rate distortion function Rµ pǫq (for its definition see Section 2). Let MT pX q be the space of T -invariant probability measures on X . They proved the following result. Theorem 1. Suppose pX , dq satisfy Condition 1.2 in [LT]. Then mdimpX , d, T q “ lim sup ǫÑ0

supµPMT pX q Rµ pǫq . | log ǫ|

For the meaning of Condition 1.2 see Definition 8. In this paper we will prove an analog formula where instead of the rate distortion function we use some terms that appear in Katok’s entropy formula. This has the advantage to simplify some computations and to make them, in the authors opinion, more intuitive. For µ P MT pX q define Nµ pn, ǫ, δq as the minimum number of pn, ǫq-dynamical balls needed to cover a set of measure strictly bigger than 1 ´ δ. Then define hµ pǫ, T, δq “ lim supnÑ8 n1 log Nµ pn, ǫ, δq. It was proven by A. Katok in [Ka] that hµ pT q “ limǫÑ0 hµ pǫ, T, δq for every ergodic measure µ and any δ P p0, 1q, where hµ pT q is the measure theoretic entropy of µ. One of the main results of this paper is the following theorem. Theorem 2. Let pX , dq be a compact metric space and T : X Ñ X continuous. Then supµPMT pX q hµ pǫ, T, δq . mdimpX , d, T q “ lim lim sup δÑ0 ǫÑ0 | log ǫ| From the proof of Theorem 2 we also recover a proof of the standard variational principle. An easy application of Theorem 2 is the computation mdimppr0, 1snqZ , dT , T q “ n,

ř 1 where the transformation T is the shift map and dT px, yq “ kPZ 2|k| dpxk , yk q, where x “ p..., x´1 , x0 , x1 ...q, y “ p..., y´1 , y0 , y1 ...q and d is the standard metric on r0, 1sn. We also investigate the connection between the rate distortion function Rµ pǫq and our replacement hµ pǫ, T, δq. For reasons that will be clear to the reader the function Rµ pǫq is closely related to r hµ pǫ, T, δq, where instead of using pn, ǫqdynamical balls we use pn, ǫq-average dynamical balls. We are in particular able to reprove Theorem 1. We also prove that the rate distortion function recovers the measure theoretic entropy in the ergodic case. Theorem 3. Let pX , dq be a compact metric space and µ P MT pX q an ergodic measure. Then hµ pT q “ lim Rµ pǫq “ r hµ pT, δq. ǫÑ0

We then obtain a result analog to the one in Theorem 2 when we replace the pn, ǫq-dynamical balls with the pn, ǫq-average dynamical balls, i.e. when replacing hµ pǫ, T, δq by r hµ pǫ, T, δq. We use this formula to reprove Theorem 1.

RATE DISTORTION THEORY, METRIC MEAN DIMENSION AND MEASURE THEORETIC ENTROPY 3

The paper is organized as follows. In Section 2 we recall some basic definitions from ergodic and information theory. In Section 3 we prove Theorem 2 and we compute the metric mean dimension of the shift over pr0, 1sn qZ . In Section 3 we also obtain a proof of the standard variational principle. In Section 4 we discuss the connections between the rate distortion function Rµ pǫq and the function r hµ pǫ, T, δq. We use the results in Section 4 to reprove Theorem 1. In Section 5 we discuss how generic is for a continuous map or homeomorphism on a manifold to have positive or maximal metric mean dimension. In Section 6 we make some final remarks and state some natural questions, we also suggest the definition of what should be the metric mean dimension of a measure. Acknowledgements. The authors would like to thank G. Iommi and M. Tsukamoto for their interest in this work. The first author would like to thanks to his advisor G. Tian for his constant support and encouragements. The last part of this paper was written when the first author was visiting Pontificia Universidad Cat´ olica de Chile, he is very grateful to G. Iommi for the invitation. Finally, the second author would like to thanks to his advisor J. Bochi for his continued guidance and encouragements. 2. Preliminaries Let pX , dq be a metric space and T : X Ñ X a continuous map. We refer to the triple pX , d, T q a dynamical system. We will not always assume that X is compact, we will specify when that assumption is required. The metric d induces a topology on X and this topology endows X with the borelian σ-algebra. Any measure on X is assumed to be defined on the borelian σ-algebra of pX , dq. We will use standard concepts in ergodic theory, for completeness we briefly define the notions more relevants to this work. A probability measure µ on X is said to be invariant under T or T -invariant if for any measurable set A we have µpAq “ µpT ´1 Aq. A probability T -invariant measure µ is said to be ergodic if any measurable set A satisfying T ´1 A “ A has measure zero or one. To a partition P “ tP1 , ..., Pm u of X we can assign the value m ÿ µpPi q log µpPi q. Hµ pPq “ ´ i“1

Observe that everytime we have a measurable map Z : X Ñ W with finite image we can associate a partition on X , the preimage partition of Z (which is a finite partition since the map has finite image). In this case we denote by Hµ pZq to the entropy of the preimage partition of Z. Let P _ Q be the refinement of the partitions P and Q. Given two random variables with finite image Z1 : X Ñ W1 and Z2 : X Ñ W2 we can define the mutual information of Z1 and Z2 as Iµ pZ1 , Z2 q “ Hµ pZ1 q ` Hµ pZ2 q ´ Hµ pZ1 , Z2 q, where Hµ pZ1 , Z2 q is the entropy of the refinement of the preimage partitions of Z1 and Z2 . We define the partitions P n :“ P _T ´1P _..._T ´pn´1qP and the quantity 1 hµ pP, T q “ inf Hµ pP n q. n n We define the entropy of the measure µ as hµ pT q “ sup hµ pP, T q, P

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where the supremum runs over finite partitions of X . For more detailed information about the entropy of a measure and its relevance in ergodic theory we refer the reader to [W]. Using the map T and the metric d we can define two new families of metrics on X by the following expressions dn px, yq “

max

kPt0,...,n´1u

dpT k x, T k yq,

n´1 1 ÿ drn px, yq “ dpT k x, T k yq. n k“0

We denote a ball of radius ǫ in the metric dn as a pn, ǫq-dynamical ball and a ball of radius ǫ in the metric drn as a pn, ǫq-average dynamical ball. The following definition allows us to state Theorem 4 in a cleaner way and also motivates Definition 2. Definition 1. Let µ be a T -invariant probability measure. For δ P p0, 1q, n P N and ǫ ą 0 we define Nµ pn, ǫ, δq to be the minimum number of pn, ǫq-dynamical balls needed to cover a set of µ-measure strictly bigger than 1 ´ δ. Set 1 hµ pǫ, T, δq “ lim sup log Nµ pn, ǫ, δq. nÑ8 n The following theorem was proven in [Ka]. Theorem 4. Let pX , dq be a compact metric space, T : X Ñ X a continuous transformation and µ an ergodic T -invariant probability measure. Then hµ pT q “ lim hµ pǫ, T, δq, ǫÑ0

where hµ pT q is the measure theoretic entropy of µ. In particular the limit above does not depend on δ P p0, 1q. Remark 1. Theorem 4 is proven by establishing inequalities between LHS and RHS. The inequality LHS ě RHS can be stated more precisely as 1 hµ pP, T q ě lim sup log Nµ pn, ǫ, δq, nÑ8 n where P is any partition of diameter less than ǫ. This fact will be used in the proof of Proposition 2. We emphasize this inequality is independient of δ. The following definition mimics Definition 1 when we use drn instead of dn . We use Katok’s formula to motivate the definition of the average measure theoretic entropy of an ergodic measure µ. Definition 2. Let µ be a T -invariant probability measure on X . We define 1 r rµ pn, ǫ, δq, hµ pǫ, T, δq “ lim sup log N nÑ8 n rµ pn, ǫ, δq is the minimum number of pn, ǫq-average dynamical where δ P p0, 1q and N balls needed to cover a set of µ-measure strictly bigger than 1 ´ δ. We define the average measure theoretic entropy of µ as the limit r hµ pǫ, T, δq. hµ pT, δq “ lim r ǫÑ0

One of the goals of this paper is to establish relations between the rate distortion function Rµ pǫq recently used by E. Lindenstrauss and M. Tsukamoto [LT] in the study of the metric mean dimension, and the quantities defined above. We start by recalling the definition of Rµ pǫq.


Definition 3. We say that the pair pX, Y q satisfies condition p˚qn,ǫ if the following properties are satisfied. (1) There exists a probability space pΩ, Pq such that X : Ω Ñ X and Y “ pY0 , ..., Yn´1 q : Ω Ñ X n are measurable functions. (2) The measure X˚ P is a T -invariant probability measure on X . řn´1 (3) Ep n1 k“0 dpT k X, Yk qq ď ǫ. We say that the pair pX, Y q satisfies condition p˚qn,ǫ,µ if moreover X˚ P “ µ. Definition 4. Given a T -invariant probability measure µ on X , define the rate distortion function as 1 Rµ pǫq “ inf IpX, Y q, n where the infimum runs over pairs pX, Y q satisfying condition p˚qn,ǫ,µ . In the definition above IpX, Y q is the mutual information of the random variables X and Y . If X and Y have finite image this was defined in the second paragraph of the introduction. For the general case (assuming X is a standard probability space) we refer the reader to Section 5.5 in [Gra]. 3. The variational principle In this section we prove Theorem 2 and the standard variational principle. Define Nd pn, ǫq as the maximal cardinality of a pn, ǫq-separated set in pX , dq and 1 SpX , d, ǫq “ lim sup log Nd pn, ǫq. nÑ8 n In case the dynamical system has been specified, we will frequently use the simplified notations Spǫq “ SpX , d, ǫq. We start with the following lemma which is important for our results. Lemma 1. Assume pX , dq is compact and T : X Ñ X a continuous map. Given ǫ ą 0, there exists a T -invariant measure µǫ such that hµǫ pǫ, T, δq ě SpX , d, 2ǫq ´ 3δSpX , d, ǫq. Proof. Let En “ tx1 , ..., xNd pn,ǫq u be a maximal collection of pn, ǫq-separated points in X . Define 1 ÿ σn “ δx , |En | xPE n

where δx is the probability measure supported at x. Then define σn “

n´1 1 ÿ k T σn . n k“0 ˚

Consider a subsequence tnk ukPN such that 1 log Nd pnk , ǫq. nk By standard arguments we can find a subsequence of tnk u, which we still denote as tnk u, such that tσ nk ukPN converges to a T -invariant probability measure µ. We can arrange the sequence such that limkÑ8 nkk “ 8. Let K be a subset of X with µpKq ą 1 ´ δ, and N pK, n, ǫq “ Nµ pn, ǫ, δq, where N pK, n, ǫq is defined to be the minimum number of pn, ǫq-dynamical balls needed to cover K. We can assume that SpX , d, ǫq “ lim

kÑ8

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K is open in X . There exists k0 such that for every k ě k0 we have σ nk pKq ą 1 ´ δ. Let Ln “ tpi, jq P N2 : 0 ď i ď n ´ 1, 1 ď j ď Nd pn, ǫqu. We assign to each point in Ln either a 0 or 1 in the following way. If T i xj P K then assign 1 to the point pi, jq, assign 0 to pi, jq otherwise. By definition of the measure σ nk we know that the number of ones in Lnk is bigger or equal than nk Nd pnk , ǫqp1 ´ δq. For s ą 1 we define Lnk psq as the set of points in Lnk with first coordinate in the interval rrnk {ks, nk ´ rnk {ss ´ 1s. The number of ones in Lnk psq is at least nk Nd pnk , ǫqp1 ´ δ ´ n1k r nsk s ´ n1k r nkk sq ě nk Nd pnk , ǫqp1 ´ δ ´ 1s ´ k1 q. 1 1 1 From now on we assume s ą 1´2δ´ 1 , in particular 1 ´ δ ´ s ´ k ą δ. We will k

1 moreover assume s ă 1´3δ , this can be done if we assume k is sufficiently large so that δ ą 1{k. We conclude that the number of ones in Lnk psq is at least nk Nd pnk , ǫqδ. Since Lnk has pnk ´ rnk {ss ´ rnk {ksq columns, there exists an index mk such that the mk th column has at least nk Nd pnk , ǫqδ{pnk ´ rnk {ss ´ rnk {ksq ones and rnk {ks ď mk ă nk ´ rnk {ss. Observe that X can be covered by at most Nd pmk , ǫ{2q pmk , ǫ{2q-dynamical balls, in particular with Nd pmk , ǫ{2q subsets of dmk -diameter smaller than ǫ. Also observe that if i ‰ j and dmk pxi , xj q ď ǫ, then dnk ´mk pT mk xi , T mk xj q ą ǫ, because dnk pxi , xj q ą ǫ by the definition of Enk . We can conclude that there exists an subset I Ă t1, ...., Nd pnk , ǫqu such that for i P I we have T mk xi P K, and |I| ě nk Nd pnk , ǫqδ{pnk ´ rnk {ss ´ rnk {ksq ą Nd pnk , ǫqδ. In other words there exists a subset A of tT mk xi uiPI such that the diameter of A with respect to dmk is at most ǫ and

|A| ě

Nd pnk , ǫqδ . Nd pmk , ǫ{2q

This implies that if a, b P A and a ‰ b, then dnk pa, bq ě dnk ´mk pa, bq ě ǫ. Then Nµ pnk , ǫ{2, δq “ N pK, nk , ǫ{2q ě

Nd pnk , ǫqδ . Nd pmk , ǫ{2q

Therefore hµ pǫ{2, T, δq “ lim sup nÑ8

1 mk 1 1 log Nd pnk , ǫq´ log Nd pmk , ǫ{2q. log Nµ pn, ǫ{2, δq ě lim sup n n n k mk k kÑ8

Recall that by construction mk ă nk ´ rnk {ss, then

mk nk

ď1´

1 s

`

1 nk .

Therefore

1 1 1 1 log Nd pnk , ǫq ´ p1 ´ ` q log Nd pmk , ǫ{2q. n s n k k mk kÑ8 1 1 1 ě lim sup log Nd pnk , ǫq ´ p3δ ` q log Nd pmk , ǫ{2q. n k mk kÑ8 nk

hµ pǫ{2, T, δq ě lim sup

Also observe that the inequality rnk {ks ď mk implies that mk Ñ 8 as k Ñ 8. By 1 log Nd pnk , ǫq, therefore construction SpX , d, ǫq “ limkÑ8 nk hµ pǫ{2, T, δq ě Spǫq ´ 3δSpǫ{2q. In particular, last lemma proves that sup hµ pǫ, T, δq ě Sp2ǫq ´ 3δSpǫq, µ


where the supremum runs over the set of T -invariant probability measures. This observation will be useful in the next corollary. A straightforward application of Lemma 1 is a proof of the standard variational principle and the proof of Theorem 2. Corollary 1 (Classical Variational principle). Let pX , dq be a compact metric space and T : X Ñ X a continuous map. Then htop pX , T q “

hµ pT q.

sup µPMT pX q

Proof. From Proposition 1 we know sup hµ pǫ, T, δq ě Sp2ǫq ´ 3δSpǫq. ş

µ

Let µ “ µx dµpxq be the ergodic decomposition of µ. The proof of the inequality in Remark 1 can be adapted to obtain hµ pǫ, T, δq ď sup hµx pQq ď xPX

hµ pT q,

sup µ ergodic

for any partition Q of diameter smaller than ǫ. Also observe that Theorem 4 implies hµ pT q ď htop pX , T q, for every ergodic measure µ. Finally Sp2ǫq ´ 3δSpǫq ď sup hµ pǫ, T, δq ď µ

sup

hµ pT q ď htop pT q.

µ ergodic

Finally taking ǫ Ñ 0 and δ Ñ 0 we get that sup

hµ pT q “ htop pT q.

µ ergodic

Corollary 2. Let pX , dq be a compact metric space and T : X Ñ X a continous map. Then supµPMT pX q hµ pǫ, T, δq mdimpX , d, T q “ lim lim sup . δÑ0 ǫÑ0 | log ǫ| Proof. The inequality LHS ě RHS follows directly from Lemma 1. The other inequality follows trivially, because by definition hµ pǫ, T, δq ď Spǫq. As an application of this formula we will compute the metric mean dimension of pr0, 1sZ, dT , T q. It is well known that mdimpr0, 1sZ, dT , T q “ 1, see for instance [LW] and [LT]. We to verify this fact. We recall that the metric dT is given by ř will proceed 1 dT px, yq “ kPZ 2|k| dpxk , yk q, where x “ p..., x´1 , x0 , x1 ...q, y “ p..., y´1 , y0 , y1 ...q and d is the standard metric on r0, 1s. Proposition 1. The metric mean dimension of the shift map on r0, 1sZ with the metric dT is given by mdimpr0, 1sZ, dT , T q “ 1. Proof. For k ě 1 consider the set Pk “ tp1 , p2 , ¨ ¨ ¨ , pk u, where pi “ 2i´1 2k . Define λk as the probability measure on r0, 1s that equidistribute the points in Pk and let µk “ pλk qbZ , the product measure on r0, 1sZ. Define Ai0 ,i2 ,...,in´1 “ tx P r0, 1sZ : x0 “ pi0 , . . . , xn´1 “ pin´1 u. We will need the following fact.

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Lemma 2. Let r ă 1{2k and q P r0, 1sZ. Then there is a unique set Ai0 ,i2 ,...,in´1 such that supppµk q X Bn pq, rq Ă Ai0 ,i2 ,...,in´1 . Proof. Let x P supppµk q X Bn pq, rq. By definition 1 , @j P t0, ¨ ¨ ¨ , n ´ 1u. 2k Since x P supppµk q we conclude xj P Pk . Otherwise the neighbourhood Nj “ ¨ ¨ ¨ ˆ r0, 1s ˆ Uj ˆ r0, 1s ˆ ¨ ¨ ¨ of x has zero µk -measure, where Uj Ă r0, 1s is an open set containing xj with Uj X Pk “ H. From the choice of r we conclude xj can take only one value in Pk , say pij . dpxj , qj q ď dpT j x, T j qq ď dn px, qq ď r ă

This lemma in particular implies that µk pBn pq, rqq ď µk pAi0 ,i2 ,...,in´1 q “

1 , kn

for every q P r0, 1sZ. Then if A satisfy µk pAq ą 1 ´ δ and A Ă 1 ´ δ ă µk pAq ď µk p

L ď

Bn pzi , rqq ď

i“1

L . kn

ŤL

i“1

Bn pzi , rq, then

This implies that Nµk pn, r, δq ě p1 ´ δqk n , for every n ě 1 and δ P p0, 1q. Finally 1 supµ hµ pr, T, δq , T, δq hµk p 3k log k ě lim ě lim “ 1. rÑ0 kÑ8 kÑ8 log 3k | log r| log 3k The opposite inequality is easier, we refer the reader to [LT] for the details.

lim

This computation trivially extends to higher dimensions. We recover the formula mdimppr0, 1sdqZ , dT , T q “ d. This can also be generalized to more general metric spaces. Let Y be a compact metric space with metric d. Given ǫ ą 0, define N pǫq as the maximal cardinality of an ǫ-separated set in pY, dq. The upper Minkowski dimension or upper box dimension of Y is defined as N pǫq dimB pY, dq “ lim sup . | log ǫ| ǫÑ0 Consider a decreasing sequence tǫk ukPN converging to zero such that limkÑ8

N pǫk q | log ǫk |

“

dimB pY q. The role of the points Pk is replaced by a maximal collection of ǫk separated points in Y . The measures λk is the probability measure that equidistributes a maximal collection of ǫk -separated points. As before we get the inequality µk pBn pq, rqq ă N pǫk q´n , for every q P Y Z and r ă ǫ2k . We finally get the inequality lim sup ǫÑ0

supµ hµ pr, T, δq log N pǫk q hµ pǫk {3, T, δq ě lim sup k ě lim “ dimB pY q. kÑ8 | log ǫk | | log r| | logpǫk {3q| kÑ8

Z , T q ě dimB pY q. Let ǫ P p0, 1q and consider a natural We conclude mdimpY ř , dT ´|j| ă ǫ{p2 ¨ diampY qq. Let M the maximum cardinality number l such that |j|ěl 2 of an ǫ-separated subset B “ txi uM i“1 of X . Consider the function defined by f pxq “ xi , where xi is the closest point to x in the subset B, whenever xi is


uniquely defined. We can extend f to a measurable function on X . Define the sets Ai “ f ´1 pxi q, and Si´l ,...,in`l “ ty P Y Z : yj P Aij for all ´ l ď j ď n ` lu, where ij P t1, ..., M u for each j. Observe that z, y P Si´l ,...,in`l implies ÿ ÿ ÿ ǫ 2´|j| ă 7ǫ, dpT i z, T i yq ď diampY q 2´|j| ` 2´|j| dpzi`j , yi`j q ă ` 2ǫ 2 |j|ěl

|j|ďl

|j|ďl

for every 0 ď i ď n. This inequality proves that every set Si´l ,...,in`l has diameter less than 7ǫ with respect to the metric dn . Since the collection of sets Si´l ,...,in`l is a covering of Y Z we conclude NdT pn, 7ǫq ď M n`2l`1 “ N pǫqn`2l`1 . Since for any measure µ we have Nµ pn, ǫ, δq ď NdT pn, ǫq, we conclude supµ hµ pǫ, T, δq lim supnÑ8 n1 log NdT pn, ǫq ď lim ǫÑ0 | log ǫ| | log ǫ| ǫÑ0 log N pǫq “ dimB pY, dq. ď lim sup | log ǫ{7| ǫÑ0 We summarize this in the following result. lim sup

Theorem 5. With the notation above we have mdimpY Z , dT , T q “ dimB pY, dq. 4. Rate distortion function and r hµ pǫ, T, δq

Proposition 2. Let µ be an ergodic T -invariant probability measure. Assume pX , dq is compact and denote its diameter by D. Then Rµ pǫq ď r hµ pǫ, T, ǫ{2Dq. Let P be any partition of diameter less than ǫ. Then Rµ pǫq ď hpP, T q. Proof. Recall that by definition Rµ pǫq is the infimum of the quantities n1 IpX, Y q, where pX, Y q satisfies condition p˚qn,ǫ,µ (here n is also allowed to vary). In particular if we want to give an upper bound to Rµ pǫq we can just exhibit a good choice of pair pX, Y q. Since pX , dq is a compact metric space it has finite diameter, call D to r pK, n, ǫq “ N rµ pn, ǫ, ǫ q “ N , the diameter of pX , dq. Choose K Ă X such that N 2D ŤN N rn pxi , ǫ{2Dq and µpKq ą i.e. there exist points txi ui“1 in X such that K Ă i“1 B ǫ 1 ´ 2D , where last union is disjoint. Pick a point p P X ztxi uN i“1 and define a measurable function f : X Ñ X in such way that for x P K we have that f pxq “ xi rn pxi , ǫ{2q, if x P X zK, then f pxq “ p. By construction for some xi with x P B |f pX q| “ N ` 1. Define Yk “ T k ˝ f ˝ X. Observe that ´ 1 n´1 ¯ ż 1 n´1 ÿ ÿ k E dpT X, Yk q “ dpT k pxq, T k f pxqqdµpxq n k“0 X n k“0 ż ż n´1 n´1 1 ÿ 1 ÿ dpT k pxq, T k f pxqqdµpxq ` dpT k pxq, T k f pxqqdµpxq “ n n X zK K k“0 k“0 ż n´1 ÿ 1 dpT k pxq, T k f pxqqdµpxq ` µpX zKqD. ď K n k“0

10


Finally observe that by definition of f and if x P K then we have drn px, f pxqq ď ǫ{2. We immediately conclude that ´ 1 n´1 ¯ ÿ E dpT k X, Yk q ď ǫ. n k“0 Since by definition of IpX, Y q we have IpX, Y q ď log |f pX q| “ logpN ` 1q, then 1 1 r pn, ǫ, ǫ{p2Dqq. Rµ pǫq “ inf IpX, Y q ď lim inf log N nÑ8 n n If we construct f using sets well approximated by pn, ǫq-dynamical balls, i.e. in the ǫ ǫ construction above we take K to satisfy N pK, n, 2D q “ Nµ pn, ǫ, 2D q, we obtain the analogous bound 1 Rµ pǫq ď lim inf log N pn, ǫ, ǫ{p2Dqq. nÑ8 n Finally use Remark 1 to conclude. The following theorem is an equivalent version of the Source Coding Theorem, which is a fundamental result in Information theory. We will use Theorem 6 in the proof of Proposition 3. For a proof and history about this theorem we refer the reader to [Gra]. Theorem 6. Let X : pΩ, Pq Ñ pX , µq be a measurable function with X˚ P “ µ, an ergodic T -invariant probability measure on X . Given ǫ0 ą 0, there exists npǫ0 q P N such that the following holds. For each n ě npǫ0 q, there exists a measurable function fn : X Ñ X n such that the pair pX, fn ˝ Xq satisfies condition p˚qn,ǫ`ǫ0 and |fn pX q| ď enpRµ pǫq`ǫ0 q . The following definition has similar content than Definition 8 but will help to simplify the language used in the proof of Proposition 3. Definition 5. A map Z is called a pn, ǫq-approximation of pX , T q if Z : X Ñ X n is a measurable map with finite image and it satisfies ż n´1 1 ÿ dpT k pxq, Zk pxqqdµpxq ď ǫ. X n k“0 Lemma 3. Given Z a pn, ǫq-approximation of pX , T q, we can find a pn, 2ǫq-approximation 1 of pX , T q, say Z 1 “ pZ01 , ..., Zn´1 q, such that Zk1 “ T k Z01 . Moreover the preimage 1 partitions of Z and Z coincide. Proof. Denote by Q “ tQ1 , ..., QT u the preimage partition of Z, this means that Z|Qj ” zj “ pzj,0 , ..., zj,n´1 q P X n . Then we have n´1 T ż ÿ 1 ÿ dpT i pxq, zk,i qdµpxq ď ǫ. n Q k i“0 k“1 řn´1 To simplify notation define gk : Qk Ñ R as gk pxq “ n1 i“0 dpT i pxq, zk,i q. Therefore the above inequality can be writen as T ż ÿ gk pxqdµpxq ď ǫ. ş

k“1 Qk

Let ak :“ Qk gk pxqdµpxq. Assume ak ‰ 0 for some k, we claim that there exists xk P Qk such that gk pxk q ď ak {µpQk q. If gk pxq ą ak {µpQk q for all x P Qk , then


integrating over Qk will lead to a contradiction by the definition of ak . If ak “ 0, we pick any element in Qk and call it xk . We define Zi1 pxq “ T i pxk q whenever x P Qk . By construction the preimage partions of Z and Z 1 coincide and ż ż n´1 n´1 1 ÿ 1 ÿ dpT i pxq, T i pxk qqdµpxq ď dpT i pxq, zk,i qdµpxq ` µpQk qgk pxk q Qk n i“0 Qk n i“0 ď 2ǫ. Proposition 3. Let µ be an ergodic T -invariant probability measure. The following inequality holds r hµ p4Lǫ, T, 1{Lq ď Rµ pǫq.

Proof. Fix ǫ0 ą 0 arbitrarily small. For n ě npǫq consider the map fn “ pfn,0 , ..., fn,n´1 q : X Ñ X n provided by Theorem 6. Since pX, fn ˝ Xq satisfies condition p˚qn,ǫ`ǫ0 we have the inequality ż n´1 1 ÿ dpT k pxq, fn,k pxqqdµpxq ď ǫ ` ǫ0 . n X k“0 In particular each map fn is a pn, ǫ ` ǫ0 q-approximation of pX , T q. Moreover 1 log |fn pX q| ďRµ pǫq ` ǫ0 . n Using Lemma 3 we find maps frn : X Ñ X n , each being a pn, 2ǫ ` 2ǫ0 qapproximation of pX , T q with the additional property that frn,k “ T k frn,0 . If we define gn :“ frn,0 , then frn “ pgn , T gn , ..., T n´1 gn q. Observe that Hµ pfn q “ Hµ pfrn q and |fn pX q| “ |frn pX q| “ |gn pX q|, in particular 1 log |gn pX q| ď Rµ pǫq ` ǫ0 . n

We have now all the ingredients to relate Rµ pǫq with the quantity r hµ p2Lǫ, T, 1{Lq. First notice that ż ż n´1 1 ÿ dpT k pxq, T k gn pxqqdµpxq ď 2ǫ ` 2ǫ0 . drn px, gn pxqqdµpxq “ X n k“0 X Ť rn pp, 2Lpǫ ` ǫ0 qq. Observe that if Define C “ gn pX q Ă X and set A “ pPC B x P X zA then drn px, gn pxqq ě 2Lpǫ ` ǫ0 q, this immediately implies µpX zAq ď 1{L or equivalently µpAq ě 1 ´ L1 . By construction the set A can be covered by |C| “ |gn pX q| pn, 2Lpǫ ` ǫ0 qq-average dynamical balls. This implies 1 rµ pn, 2Lpǫ ` ǫ0 q, 1{Lq ď 1 log |gn pX q| ď Rµ pǫq ` ǫ0 , log N n n for each n ą npǫ0 q. This implies 1 rµ pn, 2Lpǫ ` ǫ0 q, 1{Lq ď Rµ pǫq ` ǫ0 , lim sup log N nÑ8 n for each ǫ0 ą 0. Taking ǫ0 Ñ 0, 1 rµ pn, 4Lǫ, 1{Lq ď lim sup 1 log N rµ pn, 2Lpǫ ` ǫ0 q, 1{Lq ď Rµ pǫq ` ǫ0 , lim sup log N n nÑ8 n nÑ8

12


we get

1 r rµ pn, 4Lǫ, 1{Lq ď Rµ pǫq. hµ p4Lǫ, T, 1{Lq “ lim sup log N nÑ8 n

We can summarize Proposition 2 and Proposition 3 in the following result. Theorem 7. Let pX , dq be a compact metric space of diameter D and T : X Ñ X a continous transformation. Let µ be an ergodic T -invariant probability measure, then we have r hµ p4Lǫ, T, 1{Lq ď Rµ pǫq ď r hµ pǫ, T, ǫ{2Dq. Theorem 7 will be used in the proof of Theorem 3.

Definition 6. Given x P X , n ě 0 and r P p0, 1q we define 1 Bn1 px, ǫ, rq “ ty P X : #t0 ď i ď n ´ 1 : dpT i x, T i yq ď ǫu ě 1 ´ ru. n Define Nµ pn, ǫ, δ, rq as the minimum number of balls Bn1 px, ǫ, rq needed to cover a set of measure bigger or equal than δ. The following proposition follows from the main result of [ZZC]. Proposition 4. Let µ be an ergodic T -invariant probability measure. Then for each δ P p0, 1q we have the inequality 1 hµ pT q ď lim lim lim sup log Nµ pn, ǫ, δ, rq. rÑ0 ǫÑ0 nÑ8 n If we moreover assume that X is compact, then the equality holds. An easy application of Proposition 4 is the following result. Proposition 5. Let µ be an ergodic T -invariant probability measure. Then for every δ P p0, 1q we have hµ pT q ď r hµ pT, δq.

rn px, ǫq, then Proof. Observe that if y P B

#t0 ď i ď n ´ 1 : dpT i x, T i yq ą Lǫu ď n{L,

or equivalently 1 1 #t0 ď i ď n ´ 1 : dpT i x, T i yq ď Lǫu ě 1 ´ . n L rn px, ǫq Ă B 1 px, Lǫ, 1{Lq, and therefore This implies that B n rµ pn, ǫ, δq. Nµ pn, Lǫ, δ, 1{Lq ď N

By taking limits we get 1 1 rµ pn, ǫ, δq “ r lim lim sup log Nµ pn, ǫ, δ, rq ď lim lim sup log N hµ pT, δq. ǫÑ0 nÑ8 n ǫÑ0 nÑ8 n Finally using Proposition 4 we get hµ pT q ď r hµ pT, δq. We can finally prove Theorem 3.


Corollary 3. Under the hypothesis of Theorem 7 we have r hµ pT, δq “ lim Rµ pǫq “ hµ pT q ǫÑ0

Proof. Combining Theorem 7 and Proposition 2 we get r hµ pT, δq ď lim Rµ pǫq ď hµ pT q. ǫÑ0

Then Proposition 5 gives the equality of the three quantities.

We will now prove a version of the variational principle for the average dynamical distances. The proof follows closely the proof of Lemma 1 with minor modifications. rd pn, ǫq as the maximal cardinality For completeness we explain it in detail. Define N of a pn, ǫq-average separated set in pX , dq and rd pn, ǫq. r , d, ǫq “ lim sup 1 log N SpX nÑ8 n In case the dynamical system has been specified, we will frequently use the simplified r “ SpX r , d, ǫq. notation Spǫq

Lemma 4. Assume pX , dq is compact and T is injective. Given ǫ ą 0 and δ P p0, 1{4q, there exists a T -invariant probability measure µǫ such that ´ 2ǫ ¯ ´ ǫ ¯ r hµǫ pǫ, T, δq ě Sr ´ 3δ Sr . 1 ´ 4δ 1 ´ 4δ

Proof. We will fix ǫ ą 0 and δ P p0, 1{4q. Let En “ tx1 , ..., xN Ăd pn,ǫq u be a maximal collection of pn, ǫq-average separated points in X . Define 1 ÿ δx , σn “ |En | xPE n

where δx is the probability measure supported at x. Then define σn “

n´1 1 ÿ k T σn . n k“0 ˚

Consider a subsequence tnk u such that rd pnk , ǫq. r , d, ǫq “ lim 1 log N SpX kÑ8 nk By standard arguments we can find a subsequence of tnk u, that we still denote by tnk u, such that tσ nk ukPN converges to a T -invariant probability measure µ. We arrange the sequence such that limkÑ8 nkk “ 8. Let K be a subset of X with rd pK, n, p1 ´ 4δqǫ{2q “ N rµ pn, p1 ´ 4δqǫ{2, δq where N r pK, m, rq µpKq ą 1 ´ δ and N is defined to be the minimum number of pm, rq-average dynamical balls needed to cover K. We can assume that K is open in X . There exists k0 such that for every k ě k0 we have σ nk pKq ą 1 ´ δ. Let r pn, ǫqu. Ln “ tpi, jq P N2 : 0 ď i ď n ´ 1, 1 ď j ď N

We assign to each point in Ln either a 0 or 1 in the following way. If T i xj P K then assign 1 to the point pi, jq, assign 0 to pi, jq otherwise. By definition of the measure σ nk we know that the number of ones in Lnk is bigger or equal than rd pnk , ǫqp1 ´ δq. For s ą 1 we define Ln psq as the set of points in Ln with first nk N k k coordinate in the interval rrnk {ks, nk ´ rnk {ss ´ 1s. The number of ones in Lnk psq is

14


rd pnk , ǫqp1 ´ δ ´ 1 ´ 1 q. From rd pnk , ǫqp1 ´ δ ´ 1 r nk s ´ 1 r nk sq ě nk N at least nk N nk s nk k s k 1 now on we assume s ą 1´2δ´ 1 , in particular 1 ´ δ ´ 1s ´ k1 ą δ. We will moreover k

1 assume s ă 1´3δ . This can be done if we assume k is sufficiently large so that rd pnk , ǫqδ. δ ą 1{k. We conclude that the number of ones in Lnk psq is at least nk N Since Lnk has pnk ´ rnk {ss ´ rnk {ksq columns, then by the pigeonhole principle Ă

nk Nd pnk ,ǫqδ there exists an index mk such that the mk th column has at least pnk ´rn k {ss´rnk {ksq ones and rnk {ks ď mk ă nk ´ rnk {ss. Observe that X can be covered by at most rd pm, ǫ{2q pm, ǫ{2q-average dynamical balls, in particular with N rd pmk , ǫ{2q subsets N of dmk -diameter smaller than ǫ. Also observe that if i ‰ j and drmk pxi , xj q ď ǫ, then

mk ǫ `

nk ´m ÿk ´1 p“0

dpT mk `p xi , T mk `p xj q ě

mÿ k ´1

dpT p xi , T p xj q `

nÿ k ´1

dpT p xi , T p xj q

p“mk

p“0

ą nk ǫ. In the last inequality, we are using the fact that drnk pxi , xj q ą ǫ. This implies that drnk ´mk pT mk xi , T mk xj q ą ǫ. We can conclude that there exists an subrd pnk , ǫqu such that for i P I we have T mk xi P K, and |I| ě set I Ă t1, ...., N r rd pnk , ǫqδ. By the pigeonhole principle nk Nd pnk , ǫqδ{pnk ´ rnk {ss ´ rnk {ksq ą N mk there exists a subset A of tT xi uiPI such that the diameter of A with respect to drmk is at most ǫ and rd pnk , ǫqδ N |A| ě . r Nd pmk , ǫ{2q

As mentioned above this implies that if a, b P A and a ‰ b, then drnk ´mk pa, bq ě ǫ. Therefore nk ´1 ´ ´1 n k ´ mk 1 ¯ 1 ¯ drnk pa, bq ą ǫą s ǫą ´ ǫ ą 1 ´ 3δ ´ ǫ ą p1 ´ 4δqǫ. nk nk s nk nk We have assumed k is sufficiently large so that δ ą

1 nk .

Finally we obtain the bound

r rµ pnk , p1 ´ 4δqǫ{2, δq “ N rd pK, nk , p1 ´ 4δqǫ{2q ě P ě Nd pnk , ǫqδ , N rd pmk , ǫ{2q N

where P is the maximum number of pn, p1 ´ 4δqǫq-average separated points in K. Finally 1 r rµ pn, p1 ´ 4δqǫ{2, δq hµ pp1 ´ 4δqǫ{2, T, δq “ lim sup log N n nÑ8 1 rd pnk , ǫq ´ mk 1 log N rd pmk , ǫ{2q. ě lim sup log N n k mk kÑ8 nk

Recall that by construction mk ă nk ´ rnk {ss, then

mk nk

ď1´

1 s

`

1 nk .

Therefore

1 r rd pnk , ǫq ´ p1 ´ 1 ` 1 q 1 log N rd pmk , ǫ{2q. hµ pp1 ´ 4δqǫ{2, T, δq ě lim sup log N s n k mk kÑ8 nk 1 rd pnk , ǫq ´ p3δ ` 1 q 1 log N rd pmk , ǫ{2q. log N ě lim sup n n k mk k kÑ8


Also observe that the inequality rnk {ks ď mk implies that mk Ñ 8 as k Ñ 8, and rd pnk , ǫq, therefore r , d, ǫq “ limkÑ8 1 log N by construction SpX nk r r ´ 3δ Spǫ{2q. r hµ pp1 ´ 4δqǫ{2, T, δq ě Spǫq

This implies in particular that sup µPMT pX q

´ r hµ pǫ, T, δq ě Sr

´ ǫ ¯ 2ǫ ¯ ´ 3δ Sr . 1 ´ 4δ 1 ´ 4δ

As corollary of Lemma 4 we will obtain a proof of Theorem 1. For this we need a version of Theorem 6 for non-ergodic measures. To state this result we n need to introduce řn some notation. On X we consider the metric ρn given by 1 ρn px, yq “ n i“1 dpxi , yi q, where x “ px1 , ..., xn q and y “ py1 , ..., yn q. We define µn as the image of µ under the map X Ñ X n given by x şÞÑ px, T x, ..., T n´1 xq. For a finite subset Cn Ă X n we define Eµ pCn q as the integral X n mincPCn ρn px, cqdµn pxq. Definition 7. Given R ą 0 define δµ pRq “ inftEµ pCn q| Cn Ă X n and |Cn | ď enR u, and Dµ pRq “ inftEµ

¯ 1 dpT k X, Yk ˝ Xq | Iµ pX, pY0 , ..., Yn´1 qq ď Ru. n k“0 n

´ 1 n´1 ÿ

In this language the Source Coding Theorem (see Theorem 6) can be stated as Theorem 8. Let µ be an ergodic T -invariant probability measure on X . Then δµ pRq “ Dµ pRq. The following properties of δµ pRq are proven in [Gra]. Proposition 6. The function µ ÞÑ δµ pRq is affine and upper semicontinuous. The function R ÞÑ δµ pRq is convex and decreasing. As a corollary we obtain the formula ż δµx pRqdµpxq, δµ pRq “ X

ş where µ “ µx dµpxq is the ergodic decomposition of µ. It is easy to see that inf Rě0 δµ pRq “ 0. This together with the convexity of δµ pRq implies that δµ pRq is strictly decreasing at R if δµ pRq ą 0. Define D8 pRq “ sup Dµx pRq “ sup δµx pRq. x

x

Since the supremum of convex function is still convex we conclude D8 pRq is convex, in particular continuous. It follows easily from the definition that inf Rě0 D8 pRq “ 0. As before this implies that D8 pRq is strictly decreasing at R if D8 pRq ą 0. We remark that Rµ pǫq is the ‘formal’ inverse of Dµ pRq, with this we mean it is the inverse in the region where the functions are strictly decreasing. For the next remark we will use the following fact.

16


Lemma 5. Let Dµx : p0, 8q Ñ R be a convex decreasing function for each x P X and define D8 pRq “ sup Dµx pRq. x

´1 Let I “ tt P p0, 8q : D8 ptq ą 0u and J “ ImpD8 q. Then D8 : J Ñ I is well defined and for every ǫ P J we have ´1 pǫq. D8 pǫq “ sup Dµ´1 x x

Proof. As mentioned above the convexity of Dµx implies the convexity and therefore the continuity of D8 . We also know that D8 is strictly decreasing at R if D8 pRq ą ´1 0. This implies that D8 : J Ñ I is well defined and J is a connected open interval. ´1 pǫq follows from the decreasing assumption. For The inequality D8 pǫq ě supx Dµ´1 x the other inequality we argue by contradiction, i.e. we assume there exists ǫ P J ´1 pǫq pǫq, whenever Dµ´1 and ǫ0 ą 0 such that for all x P X we have D8 pǫq ´ ǫ0 ą Dµ´1 x x ´1 ´1 is well defined. Define b “ D8 pǫq and a “ D8 pǫq ´ ǫ0 {2. By the definition of D8 there exists x such that D8 paq ě Dµx paq ą ǫ “ D8 pbq ě Dµx pbq. By the intermediate value theorem exists c P ra, bq such that Dµx pcq “ ǫ. This gives a ´1 contradiction since c ą D8 pǫq ´ ǫ0 . Remark 2. Given R ą 0 and ǫ0 ą 0, there exists npǫ0 , Rq P N such that the following holds.ş For n ě npǫ0 , Rq there exists Cn Ă X n such that |Cn | ď enpR`ǫ0 q and Eµ pCn q ď Dµx pRqdµpxq ` ǫ0 . In particular Eµ pCn q ď D8 pRq ` ǫ0 . If R8 pǫq is the inverse of D8 pRq, then by Lemma 5 we know R8 pǫq “ supµx Rµx pǫq. In particular for R “ R8 pǫq and n ě npǫ0 , Rq we get a code Cn Ă X n such that |Cn | ď enpR8 pǫq`ǫ0 q and Eµ pCn q ď ǫ ` ǫ0 . Following the proof of Proposition 3 we get r hµ p4Lǫ, T, 1{Lq ď sup Rµx pǫq ď sup Rµ pǫq. x

µ ergodic

Before reproving Theorem 1 we are going to state Condition 1.2 introduced in [LT] and mentioned in the statement of Theorem 1.

Definition 8. Let pX , dq be a compact metric space. It satisfy the Condition 1.2, if for every δ ą 0, we have lim ǫδ log #pX , d, ǫq “ 0,

ǫÑ0

where #pX , d, ǫq is the minimal number of ǫ-balls needed to cover X . Remark 3. It is important to mention that every compact metrizable space admits a distance satisfying the Condition 1.2. This remark corresponds to Lemma 1.3 in [LT]. It is also important to mention that Condition 1.2 implies that r , d, ǫq SpX “ mdimpX , d, T q. lim sup | log ǫ| ǫÑ0 Proposition 7. Let pX , dq be a compact metric space and T : X Ñ X a continuous map. Then r , d, ǫq supµ Rµ pǫq SpX lim sup “ lim sup . | log ǫ| | log ǫ| ǫÑ0 ǫÑ0 If pX , dq satisfy Condition 1.2 in [LT], then mdimpX , d, T q “ lim sup ǫÑ0

supµ Rµ pǫq . | log ǫ|


Proof. The inequality lim sup ǫÑ0

r , d, ǫq supµ Rµ pǫq SpX ď lim sup , | log ǫ| | log ǫ| ǫÑ0

is the easy part of the statement. We refer the reader to [LT] for a proof. We will prove here the reversed inequality. Observe that by Remark 2 we have

In particular

r hµ p4Lǫ, T, 1{Lq ď r hµ p4ǫ{δ, T, δq ď

This implies sup µ ergodic

sup

Rµ pǫq.

µ ergodic

´ hµ p4ǫ{δ, T, δq ě Sr Rµ pǫq ě sup r µ

Rµ pǫq.

sup µ ergodic

¯ ´ ¯ 8ǫ 4ǫ ´ 3δ Sr . δp1 ´ 4δq δp1 ´ 4δq

Finally dividing by | log ǫ|, taking limsup in ǫ and then δ Ñ 0 we obtain the lower bound r supµ ergodic Rµ pǫq Spǫq ě lim sup . lim sup | log ǫ| | log ǫ| ǫÑ0 ǫÑ0 Combining Proposition 7 and Remark 3 we obtain a proof of Theorem 1. 5. Dynamics on manifolds We start this section with the following observation. Remark 4. Let F : X Ñ X be a dynamical system. The map x ÞÑ px, F x, F 2 x, ...q embedds pX , F q into pX N , T q, where T is the shift map. In other words we have a T -invariant subset Y Ă X N where pY, T q is conjugate to pX , F q. The metric dT on X N restricted to Y induces a metric ř on X . We still denote this metric by dT . More explicitely we have dT px, yq “ kě0 21k dpT k x, T k yq. Since dpx, yq ď dT px, yq for all x, y P X we conclude that mdimpX , d, F q ďmdimpX , dT , F q “ mdimpY, dT , T q ďmdimpX N , dT , T q “ dimB pX , dq. In particular the dimension of X is always an upper bound for the mean dimension. A map F that realize the equality in the above inequality is said to have maximal metric mean dimension. We will construct an example of a continuous map on the interval with maximal metric mean dimension. From now on d will always stand for the euclidean distance in r0, 1s. We start with the following Lemma. Lemma 6. Let Jk “ rak´1 , ak s Ă r0, 1s and bk “ ak ´ ak´1 . Decompose Jk into bk k 2lk equal intervals tJks u2l s“1 and define ǫk “ 2lk . Suppose that f : r0, 1s Ñ r0, 1s is a s continuous map such that Jk Ă f pJk q. Then Spr0, 1s, d, ǫk q ě logplk q.

18


Proof. Define the cylinders Ai0 ,...,im´1 “ tx P Jk : f h x P Jsih ,

for every h P t0, ..., m ´ 1uu.

We will only consider those sets with all ih ’s odd. If x and y are in distinct cylinders, then dn px, yq ě ek . Moreover by the assumption Jk Ă f pJks q we know each cylinder is nonempty. This implies that the maximal number of pm, ǫk q-separated points in Jk is at least plk qm for each m ě 1. In particular Spr0, 1s, d, ǫk q ě SpJk , d, ǫk q ě log lk . Proposition 8. There exists f : r0, 1s Ñ r0, 1s continuous such that mdimpr0, 1s, d, f q “ 1. ř ř Proof. Let bk “ C{k 2 such that kě1 bk “ 1 and lk “ k k . Define ak “ 1ďsďk bs , a0 “ 0 and Jk “ rak´1 , ak s. It is easy to construct a continuous function f such that the requirements in Lemma 6 are satisfied. We can conclude that mdimpr0, 1s, d, f q “ lim sup ǫÑ0

where ǫk “

bk 2lk

“

C 2kk`2 .

Spr0, 1s, d, ǫk q log lk Spr0, 1s, d, ǫq ě lim sup ě lim sup , | log ǫ| | log ǫk | kÑ8 kÑ8 | log ǫk |

Then

mdimpr0, 1s, d, f q ě lim sup kÑ8

log k k log lk “ lim sup “ 1. k`2 {Cq | log ǫk | kÑ8 logp2k

The opposite inequality holds for any continuous map by Remark 4 since dimB pr0, 1s, dq “ 1. A similar construction allows us to prove that if f pxq “ x has infinitely many solutions, then f can be approximated in the C 0 topology by continuous functions with metric mean dimension equal to one (slightly perturb the function f nearby the fixed points by a map as in Lemma 6 using lk sufficiently large in comparison with bk ). This argument allows us to prove that the space A “ tF P Cpr0, 1sq : mdimpr0, 1s, d, F q “ 1u, where Cpr0, 1sq is the space of continuous self maps on the interval r0, 1s with the uniform topology, is dense in Cpr0, 1sq. To see this start with G P Cpr0, 1sq. By the mean value theorem there exists xG P r0, 1s such that GpxG q “ xG . Nearby xG perturb G to a function G1 with infinitely many fixed points. Then mimic the construction done in Proposition 8 as mentioned above. We summarize this in the following result. Proposition 9. Let A “ tF P Cpr0, 1sq : mdimpr0, 1s, d, F q “ 1u. Then A is a dense subset of Cpr0, 1sq.


A similar procedure can be done in a compact manifold X of dimension bigger or equal than two with a metric d induced from a Riemannian metric g. We will prove that any homeomorphism F on X can be approximated by a homeomorphism with maximal metric mean dimension under the assumption that F has a fixed point. For simplicity in the notation we will explain the procedure for a homeomorphism in a two manifold although the same argument holds in any dimension. Let p be a fixed point of F . We make an arbitrary small perturbation F1 of F in a neighborhood of p to create countably many isolated fixed points. We will perturb F1 in a sufficiently small neighborhood of each fixed point (all neighborhood will be disjoint). As done in [Yan] we can perturb F1 in a box of length bk (for arbitrarily small bk ) around the fixed point pk such that the perturbation contains a 2lk -horseshoe. Since our box is sufficiently small we can compare the metric d with the euclidean metric up to multiplication by a uniform positive constant (here we are strongly using the compactness of X ). We will explain the local picture required in the perturbation. We start with R2 endowed with the euclidean metric d0 and a square R of lenght k bk . We define a map Gk as shown in Figure 1. Let tUi u2l i“1 be vertical strips of horizontal lenght bk {p2lk q and the regions tAi u contain in Gk pRq as in Figure 1. Let Bij “ G´1 k pAi X Uj q for pi, jq a pair of odd numbers in r1, 2lk s. A1 A2 A3

A2lk ´1 A2lk

U1 U2 U3 Figure 1: Horseshoe Define the cylinders Api0 j0 q,...,pim´1 jm´1 q “ tx P R : Gsk x P Bis js

for every s P t1, ..., m ´ 1uu.

It follows from the construction that each cylinder is not empty. Moreover if x and y are in different cylinders then d0m px, yq ą ǫk , where ǫk “ bk {p4lk q. We can conclude that SpR, d0 , ǫk q ě log lk2 . We can perturb F1 nearby the points in tpk u by a map F2 that nearby pk looks like Gk . The same reasoning as in Lemma 6 gives us that mdimpX , d, F2 q ě lim sup kÑ8

log lk2 “ lim sup logp4lk {bk q kÑ8 1 `

2 log 4 log lk

`

log b´1 k log lk

.

Observe we can assume b´1 ! logk . Therefore we found a map F2 arbitrarily k close to F such that mdimpX , d, F2 q “ 2. For a general homeomorphism we can still approximate by homeomorphisms with positive upper metric mean dimension. To see this take p P X such that F k ppq is sufficiently close to p, then perturb to

20


create a fixed point using the standard closing lemma for homeomorphism. After this approximation is done we proceed as above or as done in [Yan]. The same argument can be applied to prove that any homeomorphism F with a k-periodic point can be approximated by homeomorphisms with upper metric mean dimension at least k1 dimpX q. We denote by HpX q to the space of homeomorphisms of X . We remark that under the assumptions above we know dimB pX , dq “ dimpX q. We summarize this discussion in the following result. Proposition 10. Let pX , gq be a closed Riemannian manifold with induced metric d. The set A “ tF P HpX q : mdimpX , d, F q “ dimpX qu, is dense in the subspace of homeomorphisms with a fixed point. The set B “ tF P HpX q : mdimpX , d, F q ą 0u, is dense in HpX q. 6. Final Remarks For simplicity in previous sections we worked with the upper metric dimension. It also follows from Lemma 1 that if mdimpX , d, T q ă 8, then mdimpX , d, T q “ lim lim inf δÑ0

ǫÑ0

supµ hµ pǫ, T, δq . | log ǫ|

A more symmetric formulation of the variational principle would be mdimpX , d, T q “ lim sup ǫÑ0

supδ supµ hµ pǫ, T, δq , | log ǫ|

and

supδ supµ hµ pǫ, T, δq . ǫÑ0 | log ǫ| This also follows directly from Lemma 1. Analog statements are true for the average dynamical metric and the corresponding average metric mean dimension. The computation done in Theorem 5 also shows that the metric mean dimension depends strongly on the metric. If we have two compatible metrics d1 and d2 on a topological space Y with different box dimension, then the metric mean dimension of the shift map T on Y Z will differ when considering the metric pd1 qT and pd2 qT . It is an interesting question to describe hypothesis on the dynamics that ensure the existence of a measure of ‘maximal metric mean dimension’. In our context this would mean that hµ pǫ, T, δq mdimpX, d, T q “ lim lim sup . δÑ0 ǫÑ0 | log ǫ| mdimpX , d, T q “ lim inf

In the context of [LT], this would mean mdimpX, d, T q “ lim sup ǫÑ0

Rµ pǫq . | log ǫ|

As our bounds show, there is not major difference between this two perspectives (at least under Condition 1.2 we could link this two sides by using r hµ pǫ, T, δq). The authors believe that if X is a manifold and T preserves a OU measure µ , i.e. µ has not atoms and it is positive on every open set, then µ would have this property (at


least when choosing T generic in the space of homeomorphisms that preserve µ). To motivate the importance of this situation see [OU]. We define the upper metric mean dimension of a measure µ as hµ pǫ, T, δq . mdimµ pX , d, T q “ lim lim sup δÑ0 ǫÑ0 | log ǫ| As before, we could also use the analog formulation in the context of [LT]. We remark that this quantity depends on the metric, it is not a purely measure theoretic invariant. As remarked in the introduction of [LT], it is not possible to define a meaningful invariant that only depends on measure theoretic information. Another natural question would be if the variational principle holds for this definition, in other words, if it is true that mdimpX , d, T q “ supµ mdimµ pX , d, T q. References [Gra]

R. Gray, Entropy and information theory. Second edition. Springer, New York, 2011. xxviii+409 pp. [Gro] M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps. I. Math. Phys. Anal. Geom. 2 (1999), no. 4, 323-415. [GLT] Y. Gutman, E. Lindenstrauss, M. Tsukamoto, Mean dimension of Zk -actions. Geom. Funct. Anal. 26 (2016), no. 3, 778-817. [Ka] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. ´ Hautes Etudes Sci. Publ. Math. 51 (1980) 137-173. [L] E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem. Inst. Hautes tudes Sci. Publ. Math. No. 89 (1999), 227-262. [LT] E. Lindenstrauss, M. Tsukamoto, From rate distortion theory to metric mean dimension: variational principle, https://arxiv.org/abs/1702.05722 (2017). [LW] E. Lindenstrauss, B. Weiss, Mean topological dimension. Israel J. Math. 115 (2000), 1-24. [OU] J. Oxtoby, S. Ulam, Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math. (2) 42, (1941), 874-920. [W] P. Walters, An introduction to ergodic theory. Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. ix+250 pp. [Yan] K. Yano, A remark on the topological entropy of homeomorphisms. Invent. Math. 59 (1980), no. 3, 215-220. [ZZC] X. Zhou, L. Zhou, E. Chen, Brin-Katok formula for the measure theoretic r-entropy. C. R. Math. Acad. Sci. Paris 352 (2014), no. 6, 473- 477. Princeton University, Princeton NJ 08544-1000, USA. E-mail address: [email protected] ´ ticas, Pontificia Universidad Cato ´ lica de Chile (PUC), Avenida Facultad de Matema ˜ a Mackenna 4860, Santiago, Chile Vicun E-mail address: [email protected]