Generic homeomorphisms have full metric mean dimension
aa r X i v : . [ m a t h . D S ] O c t GENERIC HOMEOMORPHISMS HAVE FULLMETRIC MEAN DIMENSION
MARIA CARVALHO, FAGNER B. RODRIGUES, AND PAULO VARANDAS
Abstract.
We prove that the upper metric mean dimension of C -generic homeomorphisms,acting on a compact smooth boundaryless manifold with dimension greater than one, coin-cides with the dimension of the manifold. In the case of continuous interval maps we alsoshow that each level set for the metric mean dimension is C -dense in the space of continuousendomorphisms of [0 ,
1] with the uniform topology. Introduction
The topological entropy is an invariant by topological conjugation and a very useful toolto either measure how chaotic is a dynamical system or to attest that two dynamics arenot conjugate. It counts, in exponential scales, the number of distinguishable orbits up toarbitrarily small errors. Clearly, on a compact metric space, a Lipschitz map has finitetopological entropy. However, if the dynamics is just continuous, the topological entropymay be infinite. Actually, K. Yano proved in [13] that, on compact smooth manifolds withdimension greater than one, the set of homeomorphisms having infinite topological entropyare C -generic. So the topological entropy is no longer an effective label to classify them.In order to obtain a new invariant for maps with infinite entropy, E. Lindenstrauss andB. Weiss introduced in [7] the notions of upper metric mean dimension and lower metricmean dimension of an endomorphism f of a metric space ( X, d ), that we will denote bymdim M ( X, f, d ) and mdim M ( X, f, d ), respectively. These are metric versions of the meandimension , a concept proposed by M. Gromov in [1] which may be viewed as a dynamicalanalogue of the topological dimension. In particular, it is known that the mean dimension ofa homeomorphism f : X → X acting on a topological space X of finite dimension is zero.An extension of this notion to Z k -actions can be found in [2]. The upper and lower metricmean dimensions, unlike Gromov’s concept, depend on the metric adopted on the space andare nonzero only if the topological entropy of the dynamics is infinite.More recently, it was proved in [11] that, on a compact manifold with dimension greaterthan one, having positive upper metric mean dimension is a C -dense property in the wholeclass of homeomorphisms. Moreover, the authors established that the set of homeomorphismswith metric mean dimension equal to the dimension of the manifold is C -dense in the set Date : October 17, 2019.2010
Mathematics Subject Classification.
Primary: 37C45, 54H20. Secondary: 37B40, 54F45.
Key words and phrases.
Metric mean dimension; Pseudo-horseshoe; Topological dynamics.The authors have been supported by CMUP (UID/MAT/00144/2019), which is funded by FCT with na-tional (MCTES) and European structural funds through the programs FEDER, under the partnership agree-ment PT2020. PV was partially supported by Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT) - Portugal, throughthe grant CEECIND/03721/2017 of the Stimulus of Scientific Employment, Individual Support 2017 Call. of all the homeomorphisms with a fixed point. Unfortunately, the previous subset is not C -dense in the space of homeomorphisms. The existence of a fixed point is crucial due to theneed of an adequate construction of separated sets using the pseudo-horseshoes introduced in[13]. If, instead, f admits a periodic point of period p >
1, then the argument of [11] ensuresthat mdim M ( f p , X, d ) = dim X hence, as mdim M ( f p , X, d ) p mdim M ( f, X, d ),mdim M ( X, f, d ) > dim Xp . (1.1)Therefore, in order to be able to consider homeomorphisms with periodic points of arbitrarilylarge periods (actually the C generic case, as proved in [3]) and still obtain mdim M ( X, f, d ) =dim X , one must compensate for the loss of metric mean dimension caused by their likely longperiods. In this work we show that for C -generic homeomorphisms, acting on compactsmooth boundaryless manifolds with dimension greater than one, not only the metric meandimension is positive but it is equal to the dimension of the manifold. Our argument grewout of the results of [3], [13] and [5], to which we refer the reader for more background.Let us be more precise. It is known, after [13, Proposition 2], that for any homeomorphism f , any scale δ > N ∈ N there exist a C -arbitrary small perturbation g of f and asuitable iterate g k which has a compact invariant subset semi-conjugate to a subshift of finitetype with N k symbols. This ensures the existence of some scale ε > N and f ) such that the largest cardinality of any ( n, ε ) separated subset of X with respect to g satisfies s ( g, n, ε ) > N n for every ε ε and all big enough n ; so h top ( g ) > log N . AlthoughYano’s strategy succeeds in producing homeomorphisms C close to f with arbitrarily largetopological entropy, it fails to bring forth any lower bound on their metric mean dimensionsince there exists no explicit relation between ε and N . To obtain better estimates than(1.1) for the metric mean dimension, we endeavored to find such a connection in R dim X ,and then forwarded the conclusions to the manifold X using the bi-Lipschitz nature of thecharts. We have had to perform several C -small perturbations along the orbit of a periodicpoint (reminding the global changes done in the proof of Pugh’s C Closing Lemma [10]) inorder to build a new version of the pseudo-horseshoes used in [13], now obliged to satisfy twoconditions: to exist in all sufficiently small scales and to exhibit the needed separation in allmoments of the construction. We will be back to this issue on Section 7.The second question we address here concerns the space of continuous endomorphisms ofthe interval [0 ,
1] with the uniform metric, denoted by C ([0 , C ([0 , β
1, the levelset of continuous maps for which the metric mean dimension exists and is equal to β is adense subset of C ([0 , Upper and lower metric mean dimension
Most of the results we will use or prove require some mild homogeneity of the space so thatlocal perturbations can be made. For simplicity we consider here only the case of smoothcompact connected manifolds. Let X be such a manifold and d be a metric compatible withthe topology on X . Given a continuous map f : X → X and a non-negative integer n , define ENERIC HOMEOMORPHISMS HAVE FULL METRIC MEAN DIMENSION 3 the dynamical metric d n : X × X → [0 , + ∞ ) by d n ( x, y ) = max n d ( x, y ) , d ( f ( x ) , f ( y )) , . . . , d ( f n − ( x ) , f n − ( y )) o and denote by B f ( x, n, ε ) the ball of radius ε around x ∈ X with respect to the metric d n . Itis not difficult to check that d n generates the same topology as d .Having fixed ε >
0, we say that a set A ⊂ X is ( n, ε )-separated by f if d n ( x, y ) > ε forevery x, y ∈ A . Denote by s ( f, n, ε ) the maximal cardinality of all ( n, ε )-separated subsets of X by f . Due to the compactness of X , the number s ( f, n, ε ) is finite for every n ∈ N and ε > Definition 2.1.
The lower metric mean dimension of ( f, X, d ) is given by mdim M ( X, f, d ) = lim inf ε → h ( f, ε ) | log ε | where h ( f, ε ) = lim sup n → + ∞ n log s ( f, n, ε ) . Similarly, the upper metric mean dimension of ( X, f, d ) is the limit mdim M ( X, f, d ) = lim sup ε → h ( f, ε ) | log ε | . The upper/lower metric mean dimensions satisfy the following properties we may summonlater:(1) If the topological entropy h top ( f ) = lim ε → h ( f, ε ) is finite (as when f is a Lipschitzmap on a compact metric space), thenmdim M ( X, f, d ) = mdim M ( X, f, d ) = 0 . (2) Given two continuous maps f : X → X and f : X → X on compact metricspaces ( X , d ) and ( X , d ), thenmdim M ( X × X , f × f , d × d ) = mdim M ( X , f , d ) + mdim M ( X , f , d ) . (3) Given a continuous map f : X → X on a compact metric space ( X, d ), the boxdimension of (
X, d ) is an upper bound for mdim M ( X, f, d ) (cf. Remark 4 of [11]).(4) Let f : X → X be a continuous map on a compact metric space ( X, d ) and k be apositive integer. The inequalitymdim M ( X, f k , d ) k mdim M ( X, f, d )is always valid (the proof is similar to the one done for the entropy in [12]). Theequality may fail (see the previous item), though it is valid whenever f is Lipschitz,in which case these values are zero for every k ∈ N .(5) For every continuous map f : X → X on a compact metric space ( X, d ), one hasmdim M (Ω( f ) , f | Ω( f ) , d ) = mdim M ( X, f, d )where Ω( f ) stands for the set of non-wandering points of f .(6) Given a continuous map f : X → X on a compact metric space X ,mdim( X, f ) mdim M ( X, f, d ) mdim M ( X, f, d )for every metric d on X compatible with the topology of X (cf. [7, Theorem 4.2]),where mdim( X, f ) stands for the mean dimension of f . The existence of such a metric M. CARVALHO, F. RODRIGUES, AND P.VARANDAS for which the first equality holds is conjectured for general maps (cf. [6]); it is knownto be valid in the case of minimal systems (cf. Theorem 4.3 in [5]).3.
Main results
Denote by Homeo(
X, d ) the set of homeomorphisms of (
X, d ). This is a complete metricspace if endowed with the metric D ( f, g ) = max x ∈ X (cid:8) d ( f ( x ) , g ( x )) , d ( f − ( x ) , g − ( x )) (cid:9) . It is known from [11] that the upper metric mean dimension of every f ∈ Homeo(
X, d )cannot be bigger than the dimension of the manifold X . Our first result states that typicalhomeomorphisms have the largest upper metric mean dimension. We note that it is not clearwhether a similar statement for the lower metric mean dimension should hold. Theorem A.
Let ( X, d ) be a compact smooth boundaryless manifold with dimension strictlygreater than one and whose topology is induced by a distance d . There exists a C -Baireresidual subset R ⊂ Homeo(
X, d ) such that mdim M ( X, f, d ) = dim X ∀ f ∈ R . Since the manifold X has finite dimension (so its Lebesgue covering dimension is also finite),mdim( X, f ) = 0 for every f ∈ Homeo(
X, d ) (cf. [7]). Moreover, one always hasmdim(
X, f ) mdim M ( X, f, d ) . Therefore, if f ∈ R then0 = mdim( X, f ) inf ρ mdim M ( X, f, ρ ) sup ρ mdim M ( X, f, ρ ) = dim X where the infimum and supremum are taken on the space of distances ρ which induce thesame topology on X as ( X, d ). Thus, generically in Homeo(
X, d ) eithermdim(
X, f ) < inf ρ mdim M ( X, f, ρ )or inf ρ mdim M ( X, f, ρ ) < sup ρ mdim M ( X, f, ρ ) . If the conjecture mentioned in [6] turns out to be true, then it is the latter inequality thatholds C -generically.The second problem we address in this paper is closely related to the previous one. Indeed,not only the largest possible value of the metric mean dimension is significant on the spaceof dynamical systems. Actually, in the case of continuous maps on [0 ,
1] with the Euclideanmetric d , each level set for the metric mean dimension is relevant since it is dense in C ([0 , Theorem B.
Let C ([0 , be the space of continuous endomorphisms of the interval ([0 , , d ) ,where d stands for the Euclidean metric. For every β ∈ [0 , there exists a dense subset D β ⊂ C ([0 , for the uniform metric such that mdim M ([0 , , f, d ) = mdim M ([0 , , f, d ) = β ∀ f ∈ D β . Moreover, C -generically in C ([0 , one has mdim M ([0 , , f, d ) = 1 . ENERIC HOMEOMORPHISMS HAVE FULL METRIC MEAN DIMENSION 5
It is natural to consider the upper metric mean dimension as a function of three variables,namely the dynamics f , the f -invariant non-empty compact set Z ⊂ X and the metric d ,and to ask whether it varies continuously. Concerning the first variable, within the space ofhomeomorphisms satisfying the assumptions of Theorem A the irregularity of the map Z mdim M ( Z, f | Z , d ), with respect to the Hausdorff metric, is a consequence of property (5) inSection 2 together with the C -general density theorem [3]. Indeed, C -generically the non-wandering set is the limit (in the Hausdoff metric) of finite unions of periodic points, on whichthe upper metric mean dimension is zero, whereas Theorem A ensures that generically theupper metric mean dimension is positive. Regarding the second variable, in the case of smoothmanifolds ( X, d ) where the C -diffeomorphisms are C -dense on the space of homeomorphisms(which is true if the dimension of the manifold X is smaller or equal to 3, cf. [9]), Theorem Aimplies that there are no continuity points of the map f mdim M ( X, f, d ). As far as weknow, the dependence on the third variable is still an open problem.4.
Absorbing disks
In this section we address some generic topological properties of homeomorphisms actingon smooth manifolds, aiming to check the existence of absorbing disks with arbitrarily smalldiameter.Following M. Hurley in [3], if the dimension of the manifold X is dim X and D dim X denotesthe closed unit ball in R dim X , call B ⊂ X a disk if it is homeomorphic to D dim X . A closedsubset K of X is called k -absorbing for a homeomorphism f of X if f k ( K ) is contained in theinterior of K , and K is said to be absorbing if it is k -absorbing for some k ∈ N . Note that if B is a k -absorbing disk, then, by Brouwer fixed point theorem, B contains a point periodicby f with period k . We say that a point P ∈ X is a periodic attracting point for f if there isa p -absorbing disk B satisfying(1) diam( f i ( B )) < diam( B ) for every 1 i p − T j > f jp ( B ) = { P } .Observe that, since f is a bijection, the last equality implies that f p ( P ) = P . We alsoremark that, given a periodic attracting point, it is possible to choose the disk B satisfying f j ( B ) ∩ B = ∅ for every 1 j < p . In the next sections we will always assume that absorbingdisks satisfy this property.Proposition 3 in [3] ensures that for every F ∈ Homeo(
X, d ) and every ε > f ∈ Homeo(
X, d ) exhibiting a periodic attracting point and such that D ( F, f ) < ε . Noticethat having a periodic attracting point is a C quasi-robust property . More precisely, for every g ∈ Homeo(
X, d ) that is C close enough to f the following conditions hold:(a) if B is a p -absorbing disk for f ∈ Homeo(
X, d ) then B is p -absorbing for g ;(b) if B is a p -absorbing disk for f ∈ Homeo(
X, d ) then for every 1 j < p the disk f i ( B ) is p -absorbing for g ;(c) for every δ > J > f Jp ( B ) has diameter smaller than δ and is a p -absorbing disk for g .Properties (a) and (b) are immediate consequences of the closeness in the uniform topologyand the compactness of B . Property (c) is due to the attracting nature of the periodic point(that is, B is a p -absorbing disk satisfying T j > f jp ( B ) = { P } ) and item (a). Unless statedotherwise, the p -absorbing disks we will use satisfy the aforementioned properties. M. CARVALHO, F. RODRIGUES, AND P.VARANDAS
Altogether this shows that having a p -absorbing disk of diameter δ is a C -open and densecondition. Therefore, taking the intersection of the sets H n = n f ∈ Homeo(
X, d ) : f has an absorbing disk with diameter at most 1 /n o we conclude that: Lemma 4.1. C -generic homeomorphisms have absorbing disks of arbitrarily small diameter. Pseudo-horseshoes
In this section we introduce the class of invariants that will play the key role in the proofof Theorem A. They will be defined first on Euclidean spaces and afterwards conveyed tomanifolds via charts.5.1.
Pseudo-horseshoes on R k . Consider in R k the norm k ( x , · · · , x k ) k := max i k | x i | . Given r > x ∈ R k , set D kr ( x ) = n y ∈ R k : k x − y k r o D kr = D kr (cid:16) (0 , . . . , (cid:17) . For 1 j k , let π j : R k → R j be the projection on the first j coordinates. Definition 5.1.
Consider r > , x = ( x , · · · , x k ) and y = ( y , · · · , y k ) in R k , and takean open set U ⊂ R k containing D kr ( x ) . Having fixed a positive integer N , we say that ahomeomorphism ϕ : U → R k has a pseudo-horseshoe of type N at scale r connecting x to y if the following conditions are satisfied: (1) ϕ ( x ) = y . (2) ϕ (cid:16) D kr ( x ) (cid:17) ⊂ int (cid:16) D k − r ( π k − ( y )) (cid:17) × R . (3) For i = 0 , , . . . , (cid:2) N (cid:3) , ϕ (cid:16) D k − r ( π k − ( x )) × n x k − r + 4 irN o(cid:17) ⊂ int (cid:16) D k − r ( π k − ( y )) (cid:17) × ( −∞ , y k − r ) . (4) For i = 0 , , . . . , (cid:2) N − (cid:3) , ϕ (cid:18) D k − r ( π k − ( x )) × n x k − r + (4 i + 2) rN o(cid:19) ⊂ int (cid:16) D k − r ( π k − ( y )) (cid:17) × ( y k + r, + ∞ ) . (5) For each i ∈ { , . . . , N − } , the intersection V i = D kr ( y ) ∩ ϕ (cid:18) D k − r ( x ) × (cid:20) x k − r + 2 irN , x k − r + (2 i + 2) rN (cid:21)(cid:19) is connected and satisfies: (a) V i ∩ ( D k − r ( y ) × {− r } ) = ∅ ; (b) V i ∩ ( D k − r ( y ) × { r } ) = ∅ ;(c) each connected component of V i ∪ ∂D kr ( y ) is simply connected. ENERIC HOMEOMORPHISMS HAVE FULL METRIC MEAN DIMENSION 7
The name pseudo-horseshoe is adequate since, when x = y , the map ϕ does admit acompact invariant subset which is semi-conjugate to a subshift of finite type (cf. [4]). Each V i is called a vertical strip of the pseudo-horseshoe ϕ , and we denote the collection of verticalstrips of ϕ by V ϕ .Notice that this definition is both topological and geometrical. Indeed, while we considerhomeomorphisms, we also assume that certain scale is preserved and identify a preferablevertical direction by means of coordinates. Definition 5.2.
Consider ε > and a homeomorphism ϕ : U → R k with a pseudo-horseshoeof type N at scale r connecting x to y . The pseudo-horseshoe is said to be ε -separating if wemay choose the collection V ϕ so that the Hausdorff distance between distinct vertical strips isbigger than ε , that is, inf {k a − b k : a ∈ V i , b ∈ V j } > ε for every i = j . Pseudo-horseshoes on manifolds.
So far, pseudo-horseshoes were defined in opensets of R k . Now we need to convey this notion to manifolds. Definition 5.3.
Let ( X, d ) be a compact smooth manifold of dimension dim X . Given f ∈ Homeo(
X, d ) and constants < α < , δ > , < ε < δ and p ∈ N , we say that f has a ( δ, ε, p, α ) - pseudo-horseshoe if we may find a pairwise disjoint family of open subsets ( U i ) i p − of X so that f ( U i ) ∩ U ( i +1)mod p = ∅ ∀ i and a collection ( φ i ) i p − of homeomorphisms φ i : D dim Xδ ⊂ R dim X → U i ⊂ M satisfying, for every i p − : (1) ( f ◦ φ i ) ( D dim Xδ ) ⊂ U ( i +1)mod p . (2) The map ψ i = φ − i +1)mod p ◦ f ◦ φ i : D dim Xδ → R dim X has a pseudo-horseshoe of type ⌊ (cid:16) ε (cid:17) α dim X ⌋ at scale δ connecting x = 0 to itself andsuch that: (a) There are families { V i,j } j and { H i,j } j of vertical and horizontal strips, respec-tively, with j ∈ { , , . . . , ⌊ (cid:16) ε (cid:17) α dim X ⌋} , such that H i,j = ψ − i (cid:0) V i,j (cid:1) . (b) For every j = j ∈ { , , . . . , ⌊ (cid:16) ε (cid:17) α dim X ⌋} we have min n inf {k a − b k : a ∈ V i,j , b ∈ V i,j } , inf {k z − w k : z ∈ H i,j , w ∈ H i,j } o > ε. Regarding the parameters ( δ, ε, p, α ) that identify the pseudo-horseshoe, we note that δ is a small scale determined by the size of the p domains and the charts so that item (1) ofDefinition 5.3 holds; ε is the scale at which a large number (which is inversely proportional to ε and involves α ) of finite orbits is separated to comply with the demand (2) of Definition 5.3;and α is conditioned by the room in the manifold needed to build the convenient amount of ε -separated points. M. CARVALHO, F. RODRIGUES, AND P.VARANDAS
Definition 5.4.
We say that f has a coherent ( δ, ε, p, α )-pseudo-horseshoe if the pseudo-horseshoe satisfies the extra condition (3) For every i p − and every j = j ∈ { , , . . . , ⌊ (cid:16) ε (cid:17) α dim X ⌋} , the horizontalstrip H i,j crosses the vertical strip V ( i +1)mod p,j .By crossing we mean that there exists a foliation of each horizontal strip H i,j ⊂ D dim Xδ bya family C i,j of continuous curves c : [0 , → H i,j such that ψ i ( c (0)) ∈ D k − δ × {− δ } and ψ i ( c (1)) ∈ D k − δ × { δ } . There are two important main features of coherent ( δ, ε, p, α )-pseudo-horseshoes. Firstly,( δ, ε, p, α )-pseudo-horseshoes associated to a homeomorphism f persist by C perturbations of f . Secondly, if the ( δ, ε, p, α )-pseudo-horseshoe is coherent and one considers the composition ψ p − ◦ · · · ◦ ψ on the suitable subdomain of D dim Xδ , containing ⌊ (cid:16) ε (cid:17) α dim X ⌋ p horizontal stripswhich are mapped onto vertical strips and are eventually ε -separated by f up to the p th iterate.In particular, any homeomorphism f which has a coherent ( δ, ε, p, α )-pseudo-horseshoe alsohas a ( p, ε )-separated set with ⌊ (cid:16) ε (cid:17) α dim X ⌋ p elements (see Figure 5.2). It is precisely this typeof characterization of the local behavior of vertical and horizontal strips in a neighborhoodof a p -periodic point we will further select that compels the main differences between ourargument and the ones used in [11, 13]. Remark 5.5.
While vertical and horizontal strips in R k can be defined in terms of Euclideancoordinates, the same notions on the manifold X are local and depend both on the dynamicsof f and the smooth charts ( φ i ) i p − . On the manifold, the Intermediate Value Theoremensures that b H i,j := φ i ( H i,j ) ⊂ U i crosses every vertical strip b V i,j := f ( b H i,j ) as well. Remark 5.6.
To estimate the metric mean dimension using local charts taking values inEuclidean coordinates, the separation scale in Euclidean coordinates (as in Definition 5.3)has to be preserved by charts. For this reason, we assume that the local charts ( φ i ) i p − are bi-Lipschitz, and thereby we require the compact manifold to be smooth.6. Separating sets
We start linking the existence of pseudo-horseshoes to the presence of big separating sets.
Proposition 6.1.
Assume that X is a smooth compact manifold. If f ∈ Homeo(
X, d ) thenthere exists C > such that, if f has a coherent ( δ, ε, p, α ) -pseudo-horseshoe, then s (cid:0) f, p ℓ, C − ε (cid:1) > (cid:16) ⌊ (cid:16) ε (cid:17) α dim X ⌋ (cid:17) p ℓ ∀ ℓ ∈ N . (6.1) Proof.
Let N = ⌊ (cid:16) ε (cid:17) α dim X ⌋ . By assumption, there are charts ( φ i ) i p − such that each ofthe maps ψ i = φ − i +1)mod p ◦ f ◦ φ i has an ε -separating pseudo-horseshoe of type N at scale δ . Moreover, the horizontal strips ( H i,j ) j =1 , ··· , N in the domain D dim Xδ of ψ i are ε -separatedand the same holds for the vertical strips ( V i,j ) j =1 , ··· , N in the image of ψ i .Define the horizontal and vertical strips, respectively, on the manifold X by b H i,j := φ i ( H i,j ) and b V i,j := f ( b H i,j ) = ( f ◦ φ i )( b H i,j ) ENERIC HOMEOMORPHISMS HAVE FULL METRIC MEAN DIMENSION 9 ≡≡ Figure 1.
Illustration of a coherent (top) and a non-coherent (bottom)pseudo-horseshoe.for 0 i p − j N . Observe that, by construction, φ − i +1)mod p (cid:16) b V i,j (cid:17) = (cid:16) φ − i +1)mod p ◦ f ◦ φ i (cid:17) ( H i,j ) = ψ i ( H i,j ) = V i,j is a vertical strip in the domain D dim Xδ of the pseudo-horseshoe ψ i . Consider also the followingnon-empty compact subsets of X : j ∈ { , · · · , N } 7→ b K ,j := b H ,j j , j ∈ { , · · · , N } 7→ b K ,j ,j := f − ( b V ,j ∩ b H ,j ) = f − ( f ( b K ,j ) ∩ b H ,j )... ... j , j , · · · , j p ∈ { , · · · , N } 7→ b K p − ,j ,j ,...,j p := f − ( p − (cid:16) f p − ( b K p − ,j ,j ,...,j p − ) ∩ b H p − ,j p (cid:17) . Taking into account that X is a smooth manifold, we may assume that all the maps { φ ± i : 0 i p − } are Lipschitz with Lipschitz constant bounded by a uniform constant C >
1. Inparticular, by item 2(b) in Definition 5.3, there exist at least N points which are ( C − ε )-separated by f in b K ,j . Claim : With the previous notation, ( j , j ) = ( J , J ) x ∈ b K ,j ,j ⇒ x and y are (2 , C − ε ) -separated .y ∈ b K ,J ,J Indeed, as φ − is C -Lipschitz and j = J , then d ( x, y ) > d ( f ( x ) , f ( y )) > dist( b V ,j , b V ,J ) > C − dist( V ,j , V ,J ) > C − ε where dist( A, B ) := inf {k a − b k : a ∈ A, b ∈ B } , if A, B ⊂ R k inf { d ( a, b ) : a ∈ A, b ∈ B } , if A, B ⊂ X. On the other hand, if j = J and j = J , then f ( x ) , f ( y ) ∈ b V ,j but lie in different horizontalstrips; consequently, f ( x ) ∈ b V ,j and f ( y ) ∈ b V ,J and so d ( x, y ) > d ( f ( x ) , f ( y )) > C − dist( b V ,j , b V ,J ) > C − ε. Recall that we have associated to ( j , j , . . . , j p ) ∈ { , , . . . , N } p the non-empty compactset b K p − ,j ,j ,...,j p = f − ( p − (cid:16) f p − ( b K p − ,j ,j ,...,j p − ) ∩ b H p − ,j p (cid:17) and observe that, whenever ( j , j , . . . , j p ) = ( J , J , . . . , J p ), one has d p ( x, y ) > C − ε ∀ x ∈ b K p − ,j ,j ,...,j p ∀ y ∈ b K p − ,J ,J ,...,J p . This proves that s (cid:0) f, p , C − ε (cid:1) > N p . To show (6.1) for ℓ ∈ N \ { } , we repeat ℓ times the previous recursive argument for theiterate f p and the sets b K p − ,j ,j ,...,j p instead of f and the sets b K ,j . (cid:3) Corollary 6.2.
Under the assumptions of Proposition 6.1 one has lim sup n → + ∞ n log s (cid:0) f, n, C − ε (cid:1) > α dim X | log ε | . (6.2) ENERIC HOMEOMORPHISMS HAVE FULL METRIC MEAN DIMENSION 11 A C -perturbation lemma along orbits We are interested in constructing coherent pseudo-horseshoes inside absorbing disks withsmall diameter. The argument depends on a finite number of C -perturbations of the initialdynamics on disjoint supports. Furthermore, the pseudo-horseshoes will be obtained insidea small neighborhood of an orbit associated to a suitable concatenation of homeomorphisms C -close to the initial dynamics.Taking into account that X is a smooth compact boundaryless manifold, we may fix afinite atlas a whose charts are bi-Lipschitz. If r a > r a in X contains a disk D dim X ( v ) ⊂ R dim X for some v ∈ R dim X . Let L >
Proposition 7.1.
Given δ > and f ∈ H , there exist p ∈ N and < δ < δ such that, forevery < ε ≪ δ and every α ∈ (0 , , we may find g ∈ Homeo(
X, d ) satisfying: (a) g has a coherent ( δ, Lε, p, α ) -pseudo-horseshoe; (b) D ( g, f ) δ .Proof. We recall from Section 4 that C generic homeomorphisms, belonging to the residualset H given by Lemma 4.1, have absorbing disks of arbitrarily small diameter which do notdisappear under small C perturbations. More precisely, given δ >
0, each f ∈ H has both a p -absorbing disk B with diameter smaller than δ , for some p ∈ N , and an open neighborhood W f in Homeo( X, d ) such that for every g ∈ W f the disk B is still p -absorbing for g . In whatfollows we will always assume that W f is inside the open ball in (Homeo( X, d ) , D ) centeredat f with diameter δ .We start fixing coordinate systems. By Brouwer’s fixed point theorem, f has a periodicpoint P of period p in B . For every 0 i p −
1, let φ i be a bi-Lipschitz chart from D dim X ⊂ R dim X onto some open neighborhood of f i ( P ) contained in the disk f i ( B ) and suchthat φ i ((0 , · · · , f i ( P ). These charts are obtained by the composition of restrictions ofthe charts of the atlas a and possible translations, which do not affect the value of L .The next step is to choose δ > C -perturbation h ∈ Homeo(
X, d ) ofthe identity whose support has diameter smaller than 3 Lδ satisfies h ◦ f ∈ W f , and so D ( h ◦ f, f ) δ . The existence of such a δ is guaranteed by the uniform continuity of f − ,since D ( h ◦ f, f ) = max x ∈ X (cid:8) D ( h ( f ( x )) , f ( x )) , D ( f − ( h − ( x )) , f − ( x )) (cid:9) . We may assume, reducing δ if necessary, that the ball B Lδ ( f i ( P )) is strictly contained in f i ( B ) for every 0 i p −
1. In fact, we may say more: the closeness in the uniform topologyassures that the ball B Lδ ( g i ◦ · · · ◦ g ( P )) is contained in f i ( B ) for every g i which is C -closeenough to f and all 0 i p − Step 1:
Let N = ⌊ (cid:16) ε (cid:17) α dim X ⌋ . Reducing δ if necessary, we may assume that the map φ − ◦ f ◦ φ : D dim X δ → D dim X is well defined, fixes the origin and is a homeomorphism onto its image. A reasoning similarto the proof of [13, Proposition 1] provides a homeomorphism ρ : D dim X → D dim X δ isotopicto the identity and such that: (1) (cid:16) ρ ◦ φ − ◦ f ◦ φ (cid:17) ( D dim Xδ ) ⊂ int( D dim X − δ ) × ( − δ, δ ).(2) For i = 0 , , . . . , (cid:2) N (cid:3)(cid:16) ρ ◦ φ − ◦ f ◦ φ (cid:17) (cid:16) , . . . , , ( − iN ) δ (cid:17) ∈ int( D dim X − δ ) × ( − δ, − δ ) . (3) For i = 0 , , . . . , (cid:2) N − (cid:3)(cid:16) ρ ◦ φ − ◦ f ◦ φ (cid:17) (cid:16) , . . . , , ( − i + 2 N ) δ (cid:17) ∈ int( D dim X − δ ) × ( δ, δ ) . By continuity of ρ , if r > (cid:16) ρ ◦ φ − ◦ f ◦ φ (cid:17) ( D dim X − r × D δ ) ⊂ int( D dim X − δ ) × ( − δ, δ ).(2’) For i = 0 , , . . . , (cid:2) N (cid:3)(cid:16) ρ ◦ φ − ◦ f ◦ φ (cid:17) ( D dim X − r × (cid:8) ( − iN ) δ (cid:9) ) ⊂ int( D dim X − δ ) × ( − δ, − δ ) . (3’) For i = 0 , , . . . , (cid:2) N − (cid:3)(cid:16) ρ ◦ φ − ◦ f ◦ φ (cid:17) ( D dim X − r × (cid:8) ( − i + 2 N ) δ (cid:9) ) ⊂ int( D dim X − δ ) × ( δ, δ ) . ρ ◦ [ φ − ◦ f ◦ φ ] D dim X δ D dim X δ Figure 2.
Illustration of the isotopy creating a pseudo-horseshoe.
ENERIC HOMEOMORPHISMS HAVE FULL METRIC MEAN DIMENSION 13
Now, properties (1’)-(3’) imply that there exists a family V = ( V i ) i N of connecteddisjoint vertical strips such that V i = (cid:16) ρ ◦ φ − ◦ f ◦ φ (cid:17) ( K i ) ⊂ D dim Xδ for some connected subset K i ⊂ D dim X − r × (cid:20) ( − iN ) δ, ( − i + 2 N ) δ (cid:21) . The isotopic perturbation ρ of the identity can be performed so that item (5) of Definition 5.1holds, and we shall assume this is the case. Making an extra C -perturbation supported in D dim Xδ , if necessary, we ensure that the vertical strips V i are ε -distant apart. This separabilityprocess is feasible because α ∈ (0 , N = ⌊ (cid:16) ε (cid:17) α dim X ⌋ < (cid:16) ε (cid:17) dim X . Let h ∈ Homeo(
X, d ) be a homeomorphism conveying ρ to a neighborhood of f ( P ) andsuch that h ( z ) := φ ◦ ρ ◦ φ − ( z ) , if z ∈ f ( φ ( D dim X δ )) z, if z / ∈ f ( φ ( D dim X δ )) . By construction, the diameter of the support of h is smaller than 3 Lδ . By the choice of δ , this ensures that the homeomorphism f = h ◦ f belongs to W f , and so D ( f , f ) δ .Moreover, in D dim X δ one has φ − ◦ f ◦ φ = φ − ◦ h ◦ f ◦ φ = ρ ◦ φ − ◦ f ◦ φ and, consequently, f has a L − ε -separated pseudo-horseshoe of type N at scale δ connecting P to f ( P ) (which may differ from f ( P )). Thus, if p = 1, the proof of Proposition 7.1 iscomplete. Step 2:
Assume now that p >
2. By construction, the homeomorphism f belongs to W f , andso f ( B ) is a p -absorbing disk for f . Now, by a translation in the charts φ and φ in R dim X ,which does not change the Lipschitz constant L , we assume without loss of generality that φ (0 , , . . . ,
0) = f ( P ) and φ (0 , , . . . ,
0) = f ( f ( P )). Therefore, (cid:16) φ ◦ f ◦ φ − (cid:17) (0 , , . . . ,
0) =(0 , , . . . , ρ : D dim X → D dim X δ and h ( z ) := φ ◦ ρ ◦ φ − ( z ) , if z ∈ f ( φ ( D dim X δ )) z, if z / ∈ f ( φ ( D dim X δ ))such that • the support of h is contained in a ball with diameter 3 Lδ centered at f ( f ( P )); • f = h ◦ f has a L − ε -separated pseudo-horseshoe of type N at scale δ connecting f ( P ) to f ( f ( P )). The support of the perturbation h is disjoint from the one of the homeomorphism h andhas diameter smaller that 3 Lδ ; thus f ∈ W f , and so D ( f , f ) δ .Let us summarize what we have obtained so far. Under the two previous perturbations wehave built a homeomorphism f ∈ W f exhibiting two pseudo-horseshoes, one connecting P to f ( P ) and another connecting f ( P ) to f ( f ( P )). Since these perturbations are performed inEuclidean coordinates (using either the charts φ i or their modifications by rigid translations,which do not change the notions of horizontal and vertical strip), and then conveyed to themanifold X using the fixed charts, we are sure that these pseudo-horseshoes are coherent. Step 3: The recursive argument.
Set f = f . Using the previous argument recursivelywe obtain homeomorphisms { f , f , f , . . . , f p − } such that f i ∈ W f , so clearly D ( f i , f ) δ for every 1 i p −
1; besides, f p − has L − ε -separated pseudo-horseshoes connecting thesuccessive points of the finite piece of the random orbit (cid:8) P, f ( P ) , ( f ◦ f )( P ) , ( f ◦ f ◦ f )( P ) , . . . , ( f p − ◦ · · · ◦ f ◦ f ◦ f )( P ) (cid:9) . If the points ( f p − ◦ · · · ◦ f ◦ f ◦ f )( P ) and P are distinct, to end the proof of Proposition 7.1we need an extra perturbation to identify them. This last perturbation is performed in theinterior of the disk B , so the resulting homeomorphism g satisfies D ( g, f p − ) δ and g = f in X \ S j p − f j ( B ). Therefore, D ( g, f ) D ( g, f p − ) + D ( f p − , f ) δ and g has a L − ε -separated pseudo-horseshoe of type N at scale δ connecting the point P to itself. (cid:3) Remark 7.2.
For the construction of the pseudo-horseshoes it is essential that α is strictlysmaller than 1. Indeed, only if 0 < α < ⌊ (cid:16) /ε (cid:17) α dim X ⌋ points that are ε -separated inside a ball with diameter 2 δ , since this obliges ε > dim X r ⌊ (cid:16) /ε (cid:17) α dim X ⌋ ε < δ or, equivalently, 0 < ε < − α √ δ .8. Proof of Theorem A
Firstly, we note that mdim M ( X, f, d ) dim X for every f ∈ Homeo(
X, d ) (cf. [11, § X, d ).Fix a strictly decreasing sequence ( ε k ) k ∈ N in the interval (0 ,
1) which converges to zero.For any α ∈ (0 ,
1) and k ∈ N , consider the C -open set O ( ε k , α ) of the homeomorphisms g ∈ Homeo(
X, d ) such that g has a coherent ( δ, Lε k , p, α )-pseudo horseshoe, for some δ > p ∈ N and L >
0. Observe that, given α ∈ (0 ,
1) and K ∈ N , the set O K ( α ) := [ k ∈ N k > K O ( ε k , α )is C -open and, by Proposition 7.1, nonempty. Besides, it is C -dense in Homeo( X, d ) sincethe residual H (cf. Lemma 4.1) is C -dense in the Baire space Homeo( X, d ) and Proposi-tion 7.1 holds for every f ∈ H . Define R := \ α ∈ (0 , ∩ Q \ K ∈ N O K ( α ) . This is a C -Baire residual subset of Homeo( X, d ) and
Lemma 8.1. mdim M ( X, g, d ) = dim X for every g ∈ R . ENERIC HOMEOMORPHISMS HAVE FULL METRIC MEAN DIMENSION 15
Proof.
Take g ∈ R . Given a rational number α ∈ (0 ,
1) and a positive integer K , thehomeomorphism g has a coherent ( δ, Lε j K , p, α )-pseudo-horseshoe for some j K > K , δ > p ∈ N and L >
0. Therefore, by Corollary 6.2,lim sup n → + ∞ n log s ( g, n, Lε j K ) > α dim X | log ε j K | for a subsequence ( ε j K ) K ∈ N of ( ε k ) k ∈ N . Thus,mdim M ( X, g, d ) > lim sup k → + ∞ lim sup n → + ∞ n log s ( g, n, Lε k ) | log ε k | > α dim X. As α ∈ (0 , ∩ Q is arbitrary, Theorem A is proved. (cid:3) Remark 8.2.
The assumption that the manifold X has no boundary is not essential. Allowingboundary points we need to alter the argument to prove Proposition 7.1 on two instances.Firstly, absorbing disks must be considered with respect to the induced topology. Secondly,the role of Brouwer fixed point theorem is transferred to the C -closing lemma, which alsoensures the existence of a periodic point. In case this periodic point lies at the boundary ofthe manifold, an additional C -arbitrarily small perturbation yields a close homeomorphismwith an interior periodic point. Accordingly, we are obliged to change the closeness estimateon the statement of Proposition 7.1, by replacing 2 δ by 3 δ .9. Proof of Theorem B
We will start constructing piecewise affine continuous models with any prescribed met-ric mean dimension. Afterwards we will prove the theorem using surgery in the space ofcontinuous maps on the interval.9.1.
Piecewise affine models.
Denote by d the Euclidean metric in [0 ,
1] and by C ([0 , ,
1] with the uniform metric. We start describingexamples in C ([0 , β ∈ [0 , Proposition 9.1.
For every β ∈ [0 , there exists a piecewise affine function f β ∈ C ([0 , such that f β (0) = 0 , f β (1) = 1 and mdim M ([0 , , f β , d ) = β .Proof. If β = 0, the assertion is trivial: take for instance f β = identity map. Now, fix β ∈ (0 , a = a − = 1 and consider a sequence ( a k ) k ∈ N of numbers in (0 ,
1) strictlydecreasing to zero. For any k >
0, consider the interval J k = [ a k +1 , a k ]denote by γ k the diameter a k − a k +1 of J k and fix a point b k +1 of the interval ( a k +2 , a k +1 ).Let G k := [ a k +2 , a k +1 ] be the closed interval gap between J k and J k +1 .On each interval G k , define f β as a continuous piecewise affine map which maps the interval[ a k +2 , a k +1 ] onto itself, fixes the boundary points and has an attracting fixed point at b k whose topological basin of attraction contains all points in the interval ( a k +2 , a k +1 ).By construction the set S k > G k is f β -invariant, restricted to which f β has zero topologicalentropy; hence this compact set will not contribute to the metric mean dimension of f β .We now define the map f β on the set S k > J k . Let ( ℓ k ) k > be a strictly increasing sequenceof positive odd integers such that ℓ >
3. Fix k > J k in ℓ k sub-intervals ( J k,i ) i ℓ k of equal size γ k /ℓ k , where γ k = a k − a k +1 . For each 0 i < ℓ k , set c k,i := a k +1 + i γ k ℓ k . Afterwards define f β ( x ) := ℓ k γ k ( x − c k,i ) + a k +1 , if x ∈ J k, i , i i k − ℓ k γ k ( x − c k,i ) + a k , if x ∈ J k, i , i i k − i k ℓ k is given by i k := j (cid:18) ℓ k γ k (cid:19) β k . (9.2) ! " i k subintervals ! " ℓ k subintervals Figure 3.
Selection procedure of piecewise linear components of f β .In rough terms, we have defined f β on each interval J k as a piecewise affine self map takingvalues on G k ∪ J k ∪ G k − in such a way that it has a metric mean dimension close to β at acertain scale. Notice that this construction is entirely analogous to the generation process ofa ( δ, ε, p, α )-pseudo-horseshoe in Section 5, taking δ = γ k , ε = γ k /ℓ k , p = 1 and α = β . Inparticular, having such a pseudo-horseshoe is a C -open condition.In the remaining sets (cid:16) [ i < ik − J k, i (cid:17) [ (cid:16) [ i k < i ℓ k J k,i (cid:17) [ (cid:16) [ i < ik − J k, i (cid:17) (9.3)we define f β as a piecewise affine map preserving the boundary points in such a way that thesets (9.3) are mapped inside the regions G k − and G k , respectively (see e.g. Figure 9.1). Byconstruction, the map f β is continuous, piecewise affine and fixes the points 0 and 1. Claim : If the sequences ( a k ) k ∈ N and ( ℓ k ) k ∈ N satisfy the additional condition a k = a k − − a k − ℓ k − ∀ k ∈ N (9.4) then mdim M ([0 , , f β , d ) = β . ENERIC HOMEOMORPHISMS HAVE FULL METRIC MEAN DIMENSION 17
Indeed, given ε > a − a ℓ , let k = k ( ε ) ∈ N be the largest positive integersuch that ε < ε k := a k − a k +1 ℓ k . Thus ε k +1 ε < ε k , and so the assumption (9.4) ensures that a k +2 = ε k +1 ε . Therefore, f β ([0 , a k +2 ]) ⊂ [0 , a k +2 ] ⊂ [0 , ε ]and, as ε < ε k , for every n ∈ N one has s ( f β , n, ε ) > s ( f β | J ∞ k , n, ε ) > s ( f β | J ∞ k , n, ε k ) = j (cid:18) ℓ k γ k (cid:19) β k n = j ε βk k n > j ε β k n (9.5)where J ∞ k := T i ≥ f − iβ ( J k ). Consequently, as n is arbitrarymdim M ([0 , , f β , d ) > β. Before proceeding, notice that the sequences ( a k ) k ∈ N and ( ℓ k ) k ∈ N may be chosen complyingwith the condition (9.4). a k +1 a k +2 • ••• a k +3 a k b k • • • • • • Figure 4.
Local construction of an attractor between two consecutive pseudo-horseshoes.On the other hand, by construction the derivative of f β at the points of the intersection J k ∩ f − β ( J k ) ∩ . . . ∩ f − ( n − β ( J k ) is constant and equal to γ k /ℓ k . Thus, this set is formedby ( γ k ℓ k ) n disjoint and equally spaced subintervals. Moreover, any such subinterval is the( n, ε k / n, ε )-dynamical ball of f β which is contained in an ( n, ε k )-dynamical ball inside J k ∩ f − β ( J k ) ∩ . . . ∩ f − ( n − β ( J k ) has diameter smaller or equal to ε ( γ k ℓ k ) n (actually equal when dynamical balls do not intersectthe boundary of the connected components of J k ∩ f − β ( J k ) ∩ . . . ∩ f − ( n − β ( J k )). This impliesin particular that s ( f β | J ∞ k , n, ε ) s ( f β | J ∞ k , n, ε k ) · ε k ( γ k ℓ k ) n ε ( γ k ℓ k ) n = s ( f β | J ∞ k , n, ε k ) · ε k ε = j ε βk k n · ε k ε (9.6)and so lim sup n → ∞ n log s ( f β | J ∞ k , n, ε ) β | log ε k | β | log ε | . Furthermore, if 1 t < k then (9.6) also implies that s ( f β | J ∞ t , n, ε ) s ( f β | J ∞ t , n, ε t ) · ε t ε which yields lim sup n → + ∞ n log s ( f β | J ∞ t , n, ε ) β | log ε t | β | log ε | . Since ε may be taken arbitrarily small, we conclude thatmdim M ([0 , , f β , d ) β. Thus, mdim M ([0 , , f β , d ) = β. This completes the proofs of the claim and of the proposition. (cid:3)
Level sets of the metric mean dimension.
Let us now show that for every β ∈ [0 , C -dense subset D β ⊂ C ([0 , M ([0 , , f, d ) = β for every f ∈ D β . When β = 0 it is enough to take D = C [0 , C -dense subset of C [0 , C interval map f one has h top ( f ) log k f ′ k ∞ < + ∞ and, consequently,mdim M ([0 , , f, d ) = 0.Fix 0 < β f ∈ C ([0 , ε > h ∈ C ([0 , D ( f, h ) < ε and mdim M ([0 , , h, d ) = β . The proof is done througha local perturbation starting at the space of C -interval maps as we will explain. Firstly, bythe denseness of the C -interval maps we may choose h ∈ C ([0 , D ( h , h ) < ε .Secondly, if P denotes a fixed point of h (which surely exists), let h ∈ C ([0 , D ( h , h ) < ε and whose set of fixed points in a small neighborhood of P consists of aninterval J centered at P . This C -perturbation can be performed in such a way that h is C at all points except, possibly, the extreme points of J . Finally, if b J ( ˜ J ( J and b J , ˜ J areintervals of diameter smaller than ε/
3, we take a C map χ such that χ ≡ b J and χ ≡ , \ ˜ J .Let T λ denote the homothety of parameter λ ∈ (0 ,
1) and | b J | stand for the diameter of theinterval b J . Since { χ, − χ } is a partition of unity, the map h := h ,β = (1 − χ ) · h + χ · T | b J | ◦ f β ◦ T | b J | − (9.7)is continuous, coincides with h on [0 , \ ˜ J and is linearly conjugate to f β on the interval b J .Moreover, by the uniform continuity of h we can choose h so that D ( h , h ) < ε provided ENERIC HOMEOMORPHISMS HAVE FULL METRIC MEAN DIMENSION 19 that b J , ˜ J are small enough. This guarantees that D ( h , h ) < ε and, since all maps in thecombination (9.7) but f β are smooth (except possibly at two points), thenmdim M ([0 , , h , d ) = mdim M ([0 , , f β , d ) = β. This ends the proof of the first part of Theorem B.
Remark 9.2.
The case β = 1 has been considered in [11, Proposition 9].Regarding the last statement of Theorem B, we might argue as in the proof of Theorem A.However, as we have established that D is C -dense in C ([0 , α = 1 was not considered in Theorem A).Take a strictly decreasing sequence ( ε k ) k ∈ N in the interval (0 ,
1) converging to zero. Given K ∈ N , consider the non-empty C -open set D K = n g ∈ C ([0 , g has a ( γ, ε k , , k > K and γ > o . Notice that D K is C -dense in C ([0 , D := \ K ∈ N D K . This is a C -Baire residual subset of C ([0 , M ([0 , , g, d ) = 1 for every g ∈ D . Indeed, given a positive integer K , such a map g has a ( γ j K , ε j K , , j K > K and γ j K >
0. Therefore, an estimate analogous to (9.5) indicates that, fora subsequence ( ε j K ) K ∈ N of ( ε k ) k ∈ N , one haslim sup n → + ∞ n log s ( g, n, ε j K ) > | log ε j K | . Thus, mdim M ([0 , , g, d ) > lim sup k → + ∞ lim sup n → + ∞ n log s ( g, n, ε k ) | log ε k | > M ([0 , , g, d ) = 1. References [1] M. Gromov.
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CMUP & Departamento de Matem´atica, Universidade do Porto, Portugal.
E-mail address : [email protected] Departamento de Matem´atica, Universidade Federal do Rio Grande do Sul, Brazil.
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