Related papers: Variational principles of metric mean dimension fo…
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…
We introduce the notion of Feldman-Katok metric mean dimensions in this note. We show metric mean dimensions defined by different metrics coincide under weak tame growth of covering numbers, and establish variational principles for…
From a geometric perspective, we employ metric mean dimension to investigate the set of generic points of invariant measures and saturated sets in infinite entropy systems. For systems with the specification property, we establish certain…
In this paper, we studied the metric mean dimension in Feldman-Katok(FK for short) metric. We introduced the notions of FK-Bowen metric mean dimension and FK-Packing metric mean dimension on subset. And we established two variational…
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…
Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of a countable sofic group on a standard probability space admitting a generating partition with finite entropy. By applying an operator algebra perspective…
Let $(X,f)$ be a dynamical system with the specification property and $\varphi$ be continuous functions. In this paper, we establish some conditional variational principles for the upper and lower Bowen/packing metric mean dimension with…
We prove a variational principle for the metric mean dimension analog to the one in [LT]. Instead of using the rate distortion function we use the function $h_\mu(\epsilon,T,\delta)$ that is closely related to the entropy $h_\mu(T)$ of…
This paper defines and discusses the dimension notion of topological slow entropy of any subset for Z^d actions. Also, the notion of measure-theoretic slow entropy for Z^d actions is presented, which is modified from Brin and Katok [2].…
In this paper, we prove variational principles between metric mean dimension and rate distortion function for countable discrete amenable group actions which extend recently results by Lindenstrauss and Tsukamoto.
This paper contributes to the mean dimension theory of dynamical systems. We introduce a new concept called mean dimension with potential and develop a variational principle for it. This is a mean dimension analogue of the theory of…
We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax…
For dynamical systems with infinite topological entropy, the classical entropy fails to quantify their complexity effectively, while the metric mean dimension provides a natural extension in this context. In this paper, we study the…
This survey is dedicated to a new direction in the theory of dynamical systems: the dynamics of metrics in measure spaces and new (catalytic) invariants of transformations with invariant measure. A space equipped with a measure and a metric…
We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphsims, which states…
In this paper, we investigate the relations between various types of topological pressures and different versions of measure-theoretical pressures. We extend Feng- Huang's variational principle for packing entropy to packing pressure and…
In this paper we introduce three notions of measure theoretical entropy of a measurable cover U in a measure theoretical dynamical system. Two of them were already introduced in [R] and the new one is defined only in the ergodic case. We…
We introduce local topological entropy $h_{\text{top}}(T, \mathcal{U})$ and two kinds of local measure-theoretic entropy $h_{\mu}^{(r)-}(T,\mathcal{U})$ and $h_{\mu}^{(r)+}(T,\mathcal{U})$ for random bundle transformations. We derive a…
In this paper, we introduce mean dimension and rate distortion dimension for $\mathbb{Z}^{k}$-actions dynamical system $(\mathcal{X},\mathbb{Z}^k,T)$. Suppose $(\mathcal{X},\mathbb{Z}^k,T)$ has the marker property. Taking these two…
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…