Related papers: Metric mean dimension of factor maps

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…

It is well-known that the relativized variational principle established by Bogenschutz and Kifer connects the fiber topological entropy and fiber measure-theoretic entropy. In context of random dynamical systems, metric mean dimension was…

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…

M. Gromov introduced the mean dimension for a continuous map in the late 1990's, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by…

Tsukamoto (2022) introduced the notion of Bedford-McMullen carpet system, a subsystem of $([0,1]^{\mathbb{N}}\times[0,1]^{\mathbb{N}},shift)$ whose metric mean dimension and mean Hausdorff dimension does not coincide in general. The aim of…

Borrowing the idea of topological pressure determining measure-theoretical entropy in topological dynamical systems, we establish a variational principle for upper metric mean dimension with potential in terms of upper measure-theoretical…

In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by…

In this manuscript, we provide a concise review of the concept of metric dimension for both deterministic as well as random graphs. Algorithms to approximate this quantity, as well as potential applications, are also reviewed. This work has…

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…

We introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence of probability spaces and a sequence of measure-preserving maps between these spaces. This notion generalizes the classical concept of metric…

In this paper, we introduce and investigate the notions of Mean Dimension and Metric Mean Dimension for generalized iterated function systems (IFS). We establish basic properties of these invariants and prove that Mean Dimension is always…

Metric mean dimension is a metric invariant of dynamical systems. It is a dynamical analogue of Minkowski dimension of metric spaces. We explain that old ideas of Bowen (1972) can be used for clarifying the local nature of metric mean…

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…

Firstly, we answer the problem 1 asked by Gutman and $\rm \acute{\ S}$piewak in \cite{gs20}, then we establish a double variational principle for mean dimension in terms of R$\bar{e}$nyi information dimension and show the order of $\sup$…

Waist inequality is a fundamental inequality in geometry and topology. We apply it to the study of entropy and mean dimension of dynamical systems. We consider equivariant continuous maps between dynamical systems and assume that the mean…

The metric dimension of a graph is the size of the smallest set of vertices whose distances distinguish all pairs of vertices in the graph. We show that this graph invariant may be calculated by an algorithm whose running time is linear in…

We consider random labelings of finite graphs conditioned on a small fixed number of peaks. We introduce a continuum framework where a combinatorial graph is associated with a metric graph and edges are identified with intervals. Next we…

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…

We study variational principles for metric mean dimension. First we prove that in the variational principle of Lindenstrauss and Tsukamoto it suffices to take supremum over ergodic measures. Second we derive a variational principle for…

We introduce the notion of dynamical metric order of a continuous map on a compact metric space, study its basic properties, and compute it for several classes of maps. This concept which is a counterpart of the metric mean dimension with…