Related papers: Metric mean dimension of factor maps
Metric mean dimension is a geometric invariant of dynamical systems with infinite topological entropy. We relate this concept with the fractal structure of the phase space and the H\"older regularity of the map. Afterwards we improve our…
We study the relation of relative topological entropy and relative mean dimension between a factor map and its induced factor map for amenable group actions. On the one hand, we prove that a factor map has zero relative topological entropy…
We introduce some notions of conditional mean dimension for a factor map between two topological dynamical systems and discuss their properties. With the help of these notions, we obtain an inequality to estimate the mean dimension of an…
In this article, we introduce a notion of relative mean metric dimension with potential for a factor map $\pi: (X,d, T)\to (Y, S)$ between two topological dynamical systems. To link it with ergodic theory, we establish four variational…
In this paper we extend the definitions of mean dimension and metric mean di-mension for non-autonomous dynamical systems. We show some properties of this extension and furthermore some applications to the mean dimension and metric mean…
This paper is devoted to the investigation of the weighted mean topological dimension in dynamical systems. We show that the weighted mean dimension is not larger than the weighted metric mean dimension, which generalizes the classical…
This paper investigates the relationship between quantization of measures and metric mean dimension of topological dynamical systems. We introduce the concept of mean quantization dimension for invariant probability measures and establish a…
The present paper aims to investigate the metric mean dimension theory of continuous flows. We introduce the notion of metric mean dimension for continuous flows to characterize the complexity of flows with infinite topological entropy. For…
In the late 1990's, M. Gromov introduced the notion of mean dimension for a continuous map, which is, as well as the topological entropy, an invariant under topological conjugacy. The concept of metric mean dimension for a dynamical system…
Let $f: M\rightarrow M$ be a continuous map on a compact metric space $M$ equipped with a fixed metric $d$, and let $\tau$ be the topology on $M$ induced by $d$. First, we will establish some fundamental properties of the mean Hausdorff…
Metric mean dimension is a dynamical counterpart of the box dimension in fractal geometry to characterize the topological complexity of infinite entropy systems. The classical variational principle states that topological entropy equals the…
The notion of metric entropy dimension is introduced to measure the complexity of entropy zero dynamical systems. For measure preserving systems, we define entropy dimension via the dimension of entropy generating sequences. This…
We establish three variational principles for the upper metric mean dimension with potential of level sets of continuous maps in terms of the entropy of partitions and Katok's entropy of the underlying system. Our results hold for dynamical…
For infinite measure-theoretic entropy systems, we introduce the notion of measure-theoretic metric mean dimension of invariant measures for different types of measure-theoretic $\epsilon$-entropies, and show that measure-theoretic metric…
We investigate the metric mean dimension of subshifts of compact type. We prove that the metric mean dimensions of a continuous map and its inverse limit coincide, generalizing Bowen's entropy formula. Building upon this result, we extend…
It is well-known that $\epsilon$-stable sets have a deep connection with the topological entropy of dynamical systems. In the present paper, we investigate the relationships of three types of upper metric mean dimensions with potential…
In this paper, we prove that for a topological dynamical system with positive mean topological dimension and marker property, it has factors of arbitrary small mean topological dimension and zero relative mean topological dimension which…
Mean dimension is a topological invariant for dynamical systems that is meaningful for systems with infinite dimension and infinite entropy. Given a $\mathbb{Z}^k$-action on a compact metric space $X$, we study the following three problems…
We prove a variational principle for the upper and lower metric mean dimension of level sets \[ \left\{x\in X: \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))=\alpha\right\} \] associated to continuous potentials $\varphi:X\to…
For a topological system with positive topological entropy, we show that the induced transformation on the set of probability measures endowed with the weak-$*$ topology has infinite topological mean dimension. We also estimate the rate of…