Related papers: Sofic conditional mean dimension, relative sofic m…
Using a local perspective, we introduce \textit{mean dimension pairs} and give sufficient conditions of when every non-trivial factor of a continuous group action of a sofic group $G$ has positive mean dimension. In addition we show that…
We undertake a study of the conditional mean dimensions for a factor map between continuous actions of a sofic group on two compact metrizable spaces. When the group is infinitely amenable, all these concepts recover as the conditional mean…
We refine two results in the paper entitled "Sofic mean dimension" by Hanfeng Li, improving two inequalities with two equalities, respectively, for sofic mean dimension of typical actions. On the one hand, we study sofic mean dimension of…
We investigate the dynamical property of the naive mean dimension for continuous actions of any countable group on compact metrizable spaces. It is shown that naive mean dimension serves as an upper bound of sofic mean dimension for actions…
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…
Recently Bingbing Liang and Hanfeng Li computed the mean dimension and metric mean dimension for algebraic actions of amenable groups. We show how to extend their computation of metric mean dimension to the case of sofic groups, provided…
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…
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…
The main purpose of this paper is to strengthen our understanding of sofic mean dimension of two typical classes of sofic group actions. First, we study finite group actions. We prove that sofic mean dimension of any amenable group action…
For a locally compact sofic group continuously acting on a compact metric space, we first study the relative sofic entropy and prove an additive inequality relating sofic entropy and relative sofic entropy. Moreover, it is shown that the…
We study dynamical systems with the property that all the nontrivial factors have infinite topological entropy (or, positive mean dimension). We establish an ``if and only if'' condition for this property among a typical class of dynamical…
Metric mean dimension is a metric-depedent quantity to characterize the topological complexity of systems with infinite topological entropy. In this paper, we investigate metric mean dimension of factor maps. (1) We introduce three types of…
We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov-Lindenstrauss-Weiss mean dimensions for actions of countable amenable groups, and are useful for…
Let $X$ be a full-shift on the alphabet $[0, 1]^a$ and let $(Y, S)$ be an arbitrary dynamical system. We prove that any equivariant continuous map from $X$ to $Y$ has conditional metric mean dimension not less than $a-\mathrm{mdim}(Y, S)$.…
In this paper, we prove the equivalence between sofic $p$-metric mean dimension and sofic metric mean dimension. This answers a question of Hayes in \cite{HB }. Furthermore, we establish the product formula for the sofic $p$-metric mean…
We study the metric mean dimension of $\Phi$-irregular set $I_{\Phi}(f)$ in dynamical systems with the shadowing property. In particular we prove that for dynamical systems with shadowing containing a chain recurrent class $Y$, the values…
For discrete measured groupoids preserving a probability measure we introduce a notion of sofic dimension that measures the asymptotic growth of the number of sofic approximations on larger and larger finite sets. In the case of groups we…
In the context of (not necessarily minimal) actions, we consider the mean diameter and use it to characterize regular factor maps. Building on this characterization, we prove that an action is diam-mean equicontinuous if and only if it is a…
We introduce the mean topological dimension for random bundle transformations, and show that continuous bundle random dynamical systems with finite topological entropy, or the small boundary property have zero mean topological dimensions.