Dendrites and measures with discrete spectrum

Foryś-Krawiec, Magdalena; Hantáková, Jana; Kupka, Jiří; Oprocha, Piotr; Roth, Samuel

doi:10.1017/etds.2021.157

Cited by 2 publications

(1 citation statement)

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“…Corollary 4.3 has been improved by el Abdalaoui and Devianne in [2] where it is shown that if f is a continuous map on a dendrite X with zero topological entropy for which E(X) is closed and countable then the Möbius function is linearly disjoint from the system (X, f ). Recently, question 1.4 was partially answered by the second named author with coauthors in [19]. In fact, they used a slightly stronger assumption that E(X) is countable to show that every invariant measure of (X, f ) has discrete spectrum in zero entropy case.…”

Section: Addendummentioning

confidence: 99%

Quasi-graphs, zero entropy and measures with discrete spectrum

Oprocha

Zhang

2022

Nonlinearity

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In this paper, we study dynamics of maps on quasi-graphs and characterise their invariant measures. In particular, we prove that every invariant measure of a 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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Section: Addendummentioning

confidence: 99%