A variational principle for the metric mean dimension of free semigroup actions

Abstract

We consider continuous free semigroup actions generated by a family $(g_y)_{y \,\in \, Y}$ of continuous endomorphisms of a compact metric space $(X,d)$ , subject to a random walk $\mathbb P_\nu =\nu ^{\mathbb N}$ defined on a shift space $Y^{\mathbb N}$ , where $(Y, d_Y)$ is a compact metric space with finite upper box dimension and $\nu $ is a Borel probability measure on Y. With the aim of elucidating the impact of the random walk on the metric mean dimension, we prove a variational principle which relates the metric mean dimension of the semigroup action with the corresponding notions for the associated skew product and the shift map $\sigma $ on $Y^{\mathbb {N}}$ , and compare them with the upper box dimension of Y. In particular, we obtain exact formulas whenever $\nu $ is homogeneous and has full support. We also discuss several examples to enlighten the roles of the homogeneity, of the support and of the upper box dimension of the measure $\nu $ , and to test the scope of our results.

A convex analysis approach to the metric mean dimension: Limits of scaled pressures and variational principles

“…It is known (cf. [27] or [5]) that mdim M ([0, 1], | • |, T ) = 1, although, for every n ∈ N, one has mdim M (J n , | • |, T |Jn ) = 0. Let us redo this computation by applying (12) in order to show that the Dirac mass δ {1} is the unique probability measure which maximizes H K δ .…”

“…g,n (x, y) = max0 ≤ j ≤ n d(g i j g i j−1 • • • g i 1 (x), g i j g i j−1 • • • g i 1 (y))which is equivalent to d. For every ϕ ∈ C 0 (X), n ∈ N and ε > 0, take the averageS(X, d, G, ϕ, ε, n) = 1 m n |g| = n S(X, d g,n , S g,n ϕ, ε)where S(X, d g,n , S g,n ϕ, ε) is as defined in(5) andS g,n ϕ(x) = n j=0 ϕ(g i j g i j−1 • • • g i 1 (x)).ThenP (X, d, G, •, ε) : ϕ → lim sup n → +∞ 1 n log S(X, d, G, ϕ, ε, n)is an ε-pressure function. This motivates the following notion of upper metric mean dimension with potential for the semigroup action S. Given ϕ ∈ C 0 (X), the upper metric mean dimension with potential, of the action S of G on X and ϕ, is given by mdim M (X, d, G, ϕ) = lim sup ε → 0 + P (X, d, G, ϕ, ε) log (1/ε) .…”

A variational principle for weighted topological pressure under -actions

“…Recently, Chen, Dou and Zheng [CDZ22] extended the Lindenstrauss-Tsukamoto variational principle (see (1.1) or [LT18, Theorem 6]) for metric mean dimension to countably amenable group actions. Carvalho, Rodrigues and Varandas [CRV22] developed the variational principle for the metric mean dimension of free semigroup actions in terms of associated skew product and the shift. However, to the authors' best knowledge, double variational principle for mean dimension of amenable group actions, especially Z k -actions, has not been studied up to now.…”

Generic homeomorphisms have full metric mean dimension

“…Actually, while C 0 -generic dynamics have infinite topological entropy [20], the metric mean dimension can be used to detect different rates of complexity at which this may grow to infinite. An extension of this notion to Z k -actions can be found in [4,5,7], and a variational principle for the metric mean dimension of free semigroup actions appeared in [1].…”