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arXiv · 1809.05617

Quasi-graphs, zero entropy and measures with discrete spectrum

Jian Li·Piotr Oprocha·Guohua Zhang
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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.

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Jian Li, Piotr Oprocha, Guohua Zhang. 2018-09-15. Quasi-graphs, zero entropy and measures with discrete spectrum. https://doi.org/10.1088/1361-6544%2Fac4b3a

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