H\"older continuous maps on the interval with positive metric mean dimension

Fix a compact metric space $X$ with finite topological dimension. Let $C^{0}(X)$ be the space of continuous maps on $X$ and $ H^{\alpha}(X)$ the space of $\alpha$-H\"older continuous maps on $X$, for $\alpha\in (0,1].$ $H^{1}(X)$ is the space of Lipschitz continuous maps on $X$. We have $$H^{1}(X)\subset H^{\beta}(X) \subset H^{\alpha}(X) \subset C^{0}(X),\quad\text{ where }0<\alpha<\beta<1.$$ It is well-known that if $\phi\in H^{1}(X)$, then $\phi$ has metric mean dimension equal to zero. On the other hand, if $X$ is a finite dimensional compact manifold, then $C^{0}(X)$ contains a residual subset whose elements have positive metric mean dimension. In this work we will prove that, for any $\alpha\in (0,1)$, there exists $\phi\in H^{\alpha}([0,1]) $ with positive metric mean dimension.


Introduction
In [6], Lindestrauss and Weiss introduced the notion of metric mean dimension for any continuous map φ : X → X, where X is a compact metric space with metric d.We will denote by mdim M (X, d, φ) and mdim M (X, d, φ), respectively, the lower and upper metric mean dimension of φ : X → X.We have mdim M (X, d, φ) ≤ mdim M (X, d, φ). ( Denote by h top (φ) the topological entropy of φ : X → X.We have if mdim M (X, d, φ) > 0, then h top (φ) = ∞.Example 4.3 proves that there exist continuous maps φ : X → X with infinite topological entropy and metric mean dimension equal to zero.
Let N be a compact Riemannian manifold with topological dimension dim(N ) = n.In [2], the authors prove that, if n ≥ 2, then the set consisting of all homeomorphism on N with upper metric mean dimension equal to n is residual in Hom(N ).This fact is proved in [1] for C 0 (N ) instead of Hom(N ).On the other hand, any Lipschitz continuous map defined on a finite dimensional compact metric space has finite entropy (see [4], Theorem 3.2.9),therefore, it has metric mean dimension equal to zero.
For any α ∈ (0, 1), we denote by H α ([0, 1]) the set consisting of α-Hölder continuous maps on the interval [0, 1].Hazard in [3] proves that there exist continuous maps on the interval with infinite entropy which are α-Hölder for any α ∈ (0, 1).However, the example constructed by Hazard has zero metric mean dimension (see Example 4.3).In this work, we will show that, for any α ∈ (0, 1), there exists a φ ∈ H α ([0, 1]) with In the next section, we will present the definition of metric mean dimension.In Section 3, we will show some results about the metric mean dimension for continuous maps with horseshoes and several examples.Although the inequality given in (1) is clear, we do not know any reference of an explicit example of a continuous map on the interval for which the inequality is strict.We will construct this kind of examples in Section 3. Furthermore, we prove that for a, b ∈ [0, 1], with a < b, the set consisting of continuous maps φ : is dense in C 0 ([0, 1]).In Section 4 we show the existence of Hölder continuous maps with positive metric mean dimension.Finally, in the last section we state some conjectures that arise from this research.

Metric mean dimension for continuous maps
Throughout this work, X will be a compact metric space endowed with a metric d and φ : X → X a continuous map.For any n ∈ N, we define the metric d any two distinct points x, y ∈ A. We denote by sep(n, φ, ε) the maximal cardinality of any (n, φ, ε)-separated subset of X.
• We say that E ⊂ X is an (n, φ, ε)-spanning set for X if for any x ∈ X there exists y ∈ E such that d n (x, y) < ε.Let span(n, φ, ε) be the minimum cardinality of any (n, φ, ε)-spanning subset of X.
Definition 2.2.The topological entropy of (X, φ, d) is defined by Definition 2.3.We define the lower metric mean dimension and the upper metric mean dimension of (X, d, φ) by We also have that Remark 2.4.Throughout the paper, by simplicity, mdim M (X, d, φ) will be denote both quantities mdim M (X, d, φ) and mdim M (X, d, φ).
Remark 2.5.Topological entropy does not depend on the metric d.However, the metric mean dimension depends on the metric (see [6]), therefore, it is not an invariant under topological conjugacy.If X = [0, 1], we consider the metric d(x, y) = |x − y|, for every x, y ∈ [0, 1].We will denote this metric by | • |.
Remark 2.6.For any continuous map φ : X → X, it is proved in [9] that where dim B (X, d) and dim B (X, d) are respectively the lower and upper box dimension of X with respect to d.
From the remark above we have that, if N is an n-dimensional compact Riemannian manifold with Riemannian metric d, then we have that (3)

Horseshoes and metric mean dimension
An s-horseshoe for φ : [0, 1] → [0, 1] is an interval J = [a, b] ⊆ [0, 1] which has a partition into s subintervals J 1 , . . ., J s , such that J • j ∩ J • i = ∅ for i = j and J ⊆ φ(J i ) for each i = 1, . . ., s (in Figure 1 we show a 3-horseshoe).The subintervals J i will be called legs of the horseshoe J and the length |J| := b − a is its size.Misiurewicz in [8], proved if φ : [0, 1] → [0, 1] is a continuous map with h top (φ) > 0, then there exist sequences of positive integers k n and s n such that, for each n, φ kn has an s n -horseshoe and For metric mean dimension, in [9], Lemma 6, it is proved that: The lemma above provides a lower bound for the upper metric mean dimension of a continuous map φ : [0, 1] → [0, 1] with a sequence of horseshoes, since, with the conditions presented, we can have that as can be seen in the proof of Proposition 8 from [9].In order to obtain the exact value of the (lower and upper) metric mean dimension of a continuous map on the interval, we must be carefully choosing both the number and the size of the legs of the horseshoes.Inspired by the examples presented in [5], in [1] we prove the next result, which, together with the Lemma 3.1, will give us examples of continuous maps on the interval with metric mean dimension equal to a fixed value (see Examples 3.4, 3.5 and 4.4).
Theorem 3.2.Suppose for each k ∈ N there exists a s k -horseshoe for φ ∈ C 0 ([0, 1]), We can rearrange the intervals and suppose that 2 ≤ s k ≤ s k+1 for each k.If each φ| I i k : In Figure 3 we present the graph of a continuous map φ : [0, 1] → [0, 1] such that each I k is an 3 k -horseshoe for φ.Note that in Theorem 3.2,i.is presented an upper bound for the lower metric mean dimension and in ii. a condition to obtain the exact value of the upper metric mean dimension for a certain class of continuous maps on the interval.
In the next proposition we show a lower bound for the lower metric mean dimension of a continuous map on the interval satisfying the conditions in Theorem 3.2.
If lim inf The second part of the proposition follows from (3.3) and Theorem 3.2.
For α ∈ (0, 1), H α ([0, 1]) will denote the space of α-Hölder continuous maps on [0, 1].C 1 ([0, 1]) and H 1 ([0, 1]) will denote respectively the space of C 1 -maps and the space of Lipschitz continuous maps on [0, 1].We have Volumen 57, Año 2023 Next, suppose that N is a compact Riemannian manifold with topological dimension equal to n and Riemannian metric d.In [1], Theorem 4.5, it is proved that the set consisting of continuous maps on N with lower and upper metric mean dimension equal to a fixed a ∈ [0, n] is dense in C 0 (N ).Furthermore, in Theorem 4.6, it is shown that the set consisting of continuous maps on N with upper metric mean dimension equal to n is residual in C 0 (N ).If n ≥ 2, in [2], Theorem A, it is proved that the set consisting of homeomorphisms on N with upper metric mean dimension equal to n is residual in the set consisting of homeomorphisms on N .It is well known that any homeomorphism on [0, 1] has zero entropy and therefore has zero metric mean dimension.
Hence, it remains to show the existence of Hölder continuous maps on finite dimensional compact manifolds.In Theorem 4.5 we will prove that there exist Hölder continuous maps on the interval with positive metric mean dimension, with certain conditions on the Hölder exponent.In Conjectures A, B and C the authors leave three problems that can be objects of future studies.
The next lemma, whose proof is straightforward and left to the reader, will be useful in order to prove that some functions are Hölder continuous.Next example, which was introduced in [3], proves that there exist continuous maps with infinite entropy and metric mean dimension equal to zero.That map is also α-Hölder for any α ∈ (0, 1).We will include the details of its construction for the sake of completeness.
Divide each interval I n into 2n + 1 sub-intervals with the same lenght, I 1 n , . . ., I 2n+1 n .For k = 1, 3, . . ., 2n + 1, let φ| I k n : I k n → I n be the unique increasing affine map from I k n onto I n and for k = 2, 4, . . ., 2n, let φ| I k n : I k n → I n be the unique decreasing affine map from I k n onto I n .Note that φ is a continuous map (φ(x) = x for any x ∈ ∂I n ) and each I n is a (2n + 1)-horseshoe (see Figure 2), therefore We will prove that φ is α-Hölder for any α ∈ (0, 1).We will consider the map ω defined in Lemma 4.1.
Case n > m + 1 : In this case we have m+1) , 2 −m ] and consider the sub-intervals Volumen 57, Año 2023 For any x ∈ I 2m+1 m+1 , we have For any x ∈ I 1 m , we have Hence, if x ∈ I 2m+1 m+1 and y ∈ I 1 m , we have and, furthermore, . .

Given that lim
Therefore, .
We recall the map presented in ( 6) has (lower and upper) metric mean dimension equal to 1.In that case, for any k ≥ 1, has a k k -horseshoe with length equal to 6 π 2 k 2 .We can prove that is not α-Hölder for none α ∈ (0, 1), because the number of legs of each horseshoe is so big, and therefore the slopes of the map on each subinterval increases very quickly.Next, we will present a path (depending of the length of the horseshoes) consisting of continuous maps such that each of them have less legs than , their metric mean dimension is equal to one, however, they are not α-Hölder for none α ∈ (0, 1).