Mobius disjointness conjecture for local dendrite maps

TL;DR

This paper proves the M"obius disjointness conjecture for various classes of maps on dendrites and graphs, including some with zero entropy, expanding the understanding of the conjecture's validity in topological dynamics.

Contribution

It establishes the conjecture for graph maps, monotone local dendrite maps, and certain dendrite maps, including some with zero entropy, which was previously unknown.

Findings

Proved M"obius disjointness for graph maps.

Established the conjecture for all monotone local dendrite maps.

Identified a transitive dendrite map with zero entropy satisfying the conjecture.

Abstract

We prove that the M\"obius disjointness conjecture holds for graph maps and for all monotone local dendrite maps. We further show that this also hold for continuous map on certain class of dendrites. Moreover, we see that there is a transitive dendrite map with zero entropy for which M\"obius disjointness holds.