Quasi-graphs, zero entropy and measures with discrete spectrum

TL;DR

This paper investigates the dynamics of maps on quasi-graphs and dendrites, establishing that zero entropy implies discrete spectrum for invariant measures and relating positive entropy to topological horseshoes.

Contribution

It proves that invariant measures with zero entropy on quasi-graphs and certain dendrites have discrete spectrum, and links positive entropy to topological horseshoes.

Findings

Invariant measures with zero entropy have discrete spectrum on quasi-graphs.

Positive topological entropy implies existence of topological horseshoes.

Invariant measures supported on certain dendrites also have discrete spectrum.

Abstract

In this paper, we study dynamics of maps on quasi-graphs characterizing their invariant measures. In particular, we prove that every invariant measure of quasi-graph map with zero topological entropy has discrete spectrum. Additionally, we obtain an analog of Llibre-Misiurewicz's result relating positive topological entropy with existence of topological horseshoes. We also study dynamics on dendrites and show that if a continuous map on a dendrite, whose set of all endpoints is closed and has only finitely many accumulation points, has zero topological entropy, then every invariant measure supported on an orbit closure has discrete spectrum.