DENSITY OF THE LEVEL SETS OF THE METRIC MEAN DIMENSION FOR HOMEOMORPHISMS

A bstract . Let N be an n -dimensional compact riemannian manifold, with n ≥ 2. In this paper, we prove that for any α ∈ [0 , n ], the set consisting of homeomorphisms on N with lower and upper metric mean dimensions equal to α is dense in Hom( N ). More generally, given α,β ∈ [0 , n ], with α ≤ β , we show the set consisting of homeomorphisms on N with lower metric mean dimension equal to α and upper metric mean dimension equal to β is dense in Hom( N ). Furthermore, we also give a proof that the set of homeomorphisms with upper metric mean dimension equal to n is residual in Hom( N ).


Introduction
In the late 1990's, M. Gromov introduced the notion of mean topological dimension for a continuous map ϕ : X → X, which is denoted by mdim(X, ϕ), where X is a compact topological space.The mean topological dimension is an invariant under conjugacy.Furthermore, this is a useful tool in order to characterize dynamical systems that can be embedded in (([0, 1] m ) Z , σ), where σ is the left shift map on ([0, 1] m ) Z (see [12], [13]).In [8], Lindenstrauss and Weiss proved that any homeomorphism ϕ : X → X that can be embedded in (([0, 1] m ) Z , σ) must satisfy that mdim(X, ϕ) ≤ m.In [6], Gutman and Tsukamoto showed that, if (X, ϕ) is a minimal system with mdim(X, ϕ) < m/2, then we can embed it in (([0, 1] m ) Z , σ).In [11], Lindenstrauss and Tsukamoto presented an example of a minimal system with mean topological dimension equal to m/2 that cannot be embedded into (([0, 1] m ) Z , σ), which show the constant m/2 is optimal.Some applications in information theory can be found in [10] and [9].
The mean topological dimension is difficult to calculate.Therefore, Lindenstrauss and Weiss in [8] introduced the notion of metric mean dimension, which is an upper bound for the mean topological dimension.The metric mean dimension is a metric-dependent quantity (this dependence is not continuous, as we can see in [3]), therefore, it is not an invariant under topological conjugacy.
Definition 1.1.The topological entropy of ϕ : X → X is defined by Definition 1.2.We define the lower metric mean dimension and the upper metric mean dimension of (X, d, ϕ) by respectively.
Remark 1.3.Throughout the paper, we will omit the underline and the overline on the notations mdim M and mdim M when the result be valid for both cases, that is, we will use mdim M for the both cases.
In recent years, the metric mean dimension has been the subject of multiple investigations, which can be verified in the bibliography of the present work.The purpose of this manuscript is to complete the research started in [5], [14] and [2], concerning to the topological properties of the level sets of the metric mean dimension map.
In [5], Theorem C, the authors proved the set consisting of continuous maps ϕ : 1]) (see also [2], Theorem 4.1).Furthermore, in Theorem A they showed if N is an n-dimensional compact riemannian manifold, with n ≥ 2, and riemannian metric d, the set of homeomorphisms ϕ : N → N such that mdim M (N, d, ϕ) = n contains a residual set in Hom(N) (see a particular case of this fact in [14], Proposition 10).Next, for any n ≥ 1, the set consisting of continuous maps ϕ : [2], Theorem 4.5, and [1], Theorem 3.6).
We consider the next level sets of the metric mean dimension for homeomorphisms: This is due to the fact that any homeomorphism on a one-dimensional compact Riemannian manifold has zero topological entropy, leading to zero metric mean dimension.Our initial result is presented in the following theorem, proved in [1], Theorem 3.6, specifically for continuous maps on the interval.Using the techniques employed to prove Theorem 1.5, we will provide a new proof of Theorem A in [5], that is: Theorem 1.6.The set H(N) = {ϕ ∈ Hom(N) : mdim M (N, d, ϕ) = n} contains a residual set in Hom(N).
Yano, in [15], defined a kind of horseshoe in order to prove the set consisting of homeomorphisms ϕ : N → N with infinite entropy is generic in Hom(N), where N is an n dimensional compact manifold, with n ≥ 2. If we want to construct a continuous map with infinite entropy we can consider an infinite sequence of horseshoes such that the number of legs is unbounded.For the metric mean dimension case, in [5] and [14] the authors used horseshoe in order to prove the set consisting of homeomorphisms ϕ : N → N with upper metric mean dimension equal to n (which is the maximal value of the metric mean dimension for any map defined on N) is generic in Hom(N).To get metric mean dimension equal to n we must construct a sequence of horseshoes such that the number of legs increases very quickly compared to the decrease in their diameters.
Estimating the precise value of the metric mean dimension for a homeomorphism, and hence to obtain a homeomorphism with metric mean dimension equal to a fixed α ∈ (0, n), is harder and not trivial sake.We need to establish a precise relation between the sizes of the horseshoes together with the number of appropriated legs to control the metric mean dimension.This is our main tool (see Lemma 2.3).
The paper is organized as follows: In Section 2, we will construct homeomorphisms, defined on an n-cube, with metric mean dimension equal to α, for a fixed α ∈ (0, n].Furthermore, given α, β ∈ [0, n], with α ≤ β, we will construct examples of homeomorphisms ϕ : In Section 3, we will prove Theorem 1.5.Finally, in Section 4, we will show Theorem 1.6.

Homeomorphisms on the n-cube with positive metric mean dimension
Let n ≥ 2. Given α, β ∈ [0, n], α ≤ β, in this section we will construct a homeomorphism ϕ α,β , defined on an n-cube, with lower metric mean dimension equal to α and upper metric mean dimension equal to β (see Lemma 2.4).This construction will be the main tool to prove the Theorem 1.5, since if a homeomorphism present a periodic orbit, then we can glue, in the C 0 -topology, along this orbit the dynamic of ϕ α,β .
The construction of ϕ α,β requires a special type of horseshoe.So, let us present the first definition.
2).Thus, our strategy to prove the Theorem 1.5 will be to use local charts to glue such horseshoes along of periodic orbits of a homeomorphism.
Inspired by the results shown in [5] and [14] to obtain a homeomorphism ϕ : N → N with upper metric mean dimension equal to dim(N), we present the next lemma, which proves for any α ∈ (0, n], there exists a homeomorphism ϕ : C := [0, 1] n → C, with (upper and lower) metric mean dimension equal to α.

Homeomorphisms on manifolds with positive metric mean dimension
Throughout this section, N will denote an n dimensional compact riemannian manifold with n ≥ 2 and d a riemannian metric on N. On Hom(N) we will consider the metric d(ϕ, φ) = max It is well-known (Hom(N), d) is a complete metric space.The main goal of this section is to prove Theorem 1.5.
Good Charts.For each p ∈ N, consider the exponential map where 0 p is the origin in the tangent space T p N, δ ′ is the injectivity radius of N and B ϵ (x) denote the open ball of radius ϵ > 0 with center x.We will take δ N = δ ′ 2 .The exponential map has the following properties: • Since N is compact, δ ′ does not depends on p.
• The derivative of exp p at the origin is the identity map: Proof.Let P r (N) be the set of C r -diffeomorphisms on N with a periodic point.This set is C 0 -dense in Hom(N) (see [4], [7]).Hence, in order to prove the theorem, it is sufficient to show that H α (N) is dense in P r (N).Fix ψ ∈ P r (N) and take any ε ∈ (0, δ N ).Suppose that p ∈ N is a periodic point of ψ, with period k.Let β > 0, small enough, such that ) has two connected components.We denote by C i the connected component that contains ψ i (p).Consider the positive number Then, there exists a homeomorphism where . Hence, D ⊆ D i is an (s, ϕ α , ϵ)-separated set if and only if exp ψ i (p) (D) ⊆ N is an (s, φ α , ϵ)separated set for any ϵ > 0. Therefore, sep(s, φ α | K , ϵ) = k sep(s, ϕ α , ϵ).Therefore, which proves the theorem.□ The last theorem proved Theorem 1.5 in the case α = β = n.The proof of the general case will be a consequence of the above arguments, in fact: Proof of the Theorem 1.5.The proof follows similarly to the proof of Theorem 3.1, changing ϕ α by a continuous map ϕ α,β : as in the Lemma 2.4.□

Tipical homeomorphism has maximal metric mean dimension
To complete this work, in this section we show Theorem 1.6, which was proved in Theorem A in [5], however, we will present an alternative proof of this fact using the techniques of Section 2 and Section 3.
Using local charts, the last definition can be done on the manifold N. Definition 4.3.Let N be an n-dimensional riemannian manifold and fix k ≥ 1.We say that ϕ ∈ Hom(N) has an n-dimensional strong (ϵ, (2k + 1) n−1 )-horseshoe E = [a, b] n , if there is s and an exponential charts exp i : B δ N (0) → N, for i = 1, . . ., s, such that: To simplify the notation, we will set ϕ i = ϕ for each i = 1, . . ., s.
For ϵ > 0 and k ∈ N, we consider the sets Lemma 4.4.The set H is residual.
Proof.Clearly, for any ϵ ∈ (0, δ N ) and k ∈ N, the set H(ϵ, k) is open and nonempty.We claim that the set H(k) is dense in Hom(N).In fact: fix ψ ∈ P r (N) with a s-periodic point.In the same way as the proof of the Theorem 3.1, every small neighborhood of the orbit of this point can be perturbed in order to obtain a strong 1 i 2 , 3 nki horseshoe for a ϕ such that ϕ 2 be close to ψ for a large enough i.Thus H(k) is a dense set, and a fortiori □ Finally, we prove Theorem 1.6.

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