Topological dynamics of Markov multi-maps of the interval

TL;DR

This paper explores the topological dynamics of Markov multi-maps on the interval, establishing connections between their properties and associated shifts of finite type, and characterizing inverse limit systems.

Contribution

It introduces a framework linking Markov multi-maps with shifts of finite type and characterizes their dynamic properties and inverse limit systems.

Findings

Properties of the associated SFT determine the topological dynamics of the multi-map.

The paper characterizes when inverse limit systems exhibit properties like transitivity and mixing.

Results extend understanding of entropy relationships in Markov multi-maps.

Abstract

We study Markov multi-maps of the interval from the point of view of topological dynamics. Specifically, we investigate whether they have various properties, including topological transitivity, topological mixing, dense periodic points, and specification. To each Markov multi-map, we associate a shift of finite type (SFT), and then our main results relate the properties of the SFT with those of the Markov multi-map. These results complement existing work showing a relationship between the topological entropy of a Markov multi-map and its associated SFT. We also characterize when the inverse limit systems associated to the Markov multi-maps have the properties mentioned above.