A large class of dendrite maps for which M\"{o}bius disjointness property of Sarnak is fulfilled

TL;DR

This paper proves that a broad class of dendrite maps with closed and countable endpoints satisfy Sarnak's M"{o}bius disjointness conjecture, extending previous results and properties to more dendrite classes.

Contribution

It extends the class of dendrite maps known to fulfill Sarnak's M"{o}bius disjointness property, including those with closed and countable endpoints.

Findings

Dendrite maps with closed, countable endpoints satisfy Sarnak's conjecture.

The Smital-Ruelle property is extended to certain dendrites.

The result generalizes previous work by el Abdalaoui-Askri and Marzougui.

Abstract

We prove that any dendrite map for which the set of endpoints is closed and countable fulfilled Sarnak M\"{o}bius disjointness. This extended a result by el Abdalaoui-Askri and Marzougui \cite{ela-GH}. We further notice that the Smital-Ruelle property can be extended to the class of dendrites with closed and countable endpoints.