Spectral decomposition and entropy for open set-valued maps

TL;DR

This paper extends the spectral decomposition framework to open set-valued maps, demonstrating their complex dynamics, including infinite entropy and links to Anosov systems, thus advancing the understanding of their structure.

Contribution

It introduces a spectral decomposition approach for open set-valued maps, incorporating transitivity and mixing, and establishes their connection to Anosov diffeomorphisms.

Findings

Open set-valued maps have infinite topological entropy.

The framework links transitive open set-valued maps to Anosov diffeomorphisms.

Spectral decomposition applies to these maps, revealing their complex structure.

Abstract

In this paper, we study the dynamics of set-valued maps whose graphs are open and such that the image of each point is an open and connected set. Building upon the work of P. Duarte and M. Torres, who introduced and analyzed the combinatorial structure of the final recurrent set, we incorporate the notions of transitivity and mixing, thereby bringing their framework into the spirit of Smale's spectral decomposition theorem. We also demonstrate that such maps exhibit infinite topological entropy, and we establish a connection between transitive open set-valued maps and transitive Anosov diffeomorphisms.