Topological entropy, mean dimension, and weakly equivalent flows

TL;DR

This paper revisits the theory of topological entropy for weakly equivalent flows, introducing a more straightforward topological approach that unifies the study of topological entropy and mean dimension, extending and refining Ohno's classical results.

Contribution

It develops a new topological method for analyzing weakly equivalent flows, unifying the treatment of topological entropy and mean dimension, and refines previous results by Ohno with a more elementary approach.

Findings

Reestablished Ohno's theorem for flows without fixed points using a new method.

Extended the theory to include mean dimension in the context of weakly equivalent flows.

Refined Ohno's example to better understand topological complexity.

Abstract

In this paper, we mainly revisit a nice theory for topological entropy of weakly equivalent flows, which was originally investigated by Ohno in 1980. We will develop a new approach, being more straightforward and elementary than the measure-theoretic one provided by Ohno, to the theory for weak equivalence of flows, and as a novelty, we study both topological entropy and mean dimension with a highly unified process in relation to such objects. In particular, for weakly equivalent flows without fixed points we recover Ohno's theorem for topological entropy relation with a substantially different method, and moreover, carry out an analogue within the framework of mean dimension; while for weakly equivalent flows with fixed points, our technique refines the procedure suggested in Ohno's construction, and strengthens Ohno's example with a view towards topological complexity of dynamical systems. Essentially, our method is topological. For this purpose, we first introduce a modification to the definition of topological entropy, by relating a spanning set inside the state space to a finite set outside, which comes to be a tiny but key difference, via a map, and further, show that it leads eventually to the same value as the topological entropy. Although to an intermediate extent, the modified quantity generally does not coincide with the one appearing in the classical definition, it has some basic combinatorial feature, which not only applies flexibly to topological entropy, but also adapts directly to the mean dimension context. Using this alternative, we are allowed to re-establish and enrich Ohno's theory.