Transitive dendrite map with zero entropy

TL;DR

This paper constructs a transitive dendrite map with zero topological entropy, demonstrating that weak mixing can occur without positive entropy, thus answering a longstanding open question.

Contribution

The authors modify a known example to produce a dendrite map that is transitive and weakly mixing yet has zero entropy, revealing new dynamics possibilities.

Findings

Existence of a transitive dendrite map with zero entropy

Weak mixing can occur without positive entropy in dendrite maps

Answers an open question in topological dynamics

Abstract

Hoehn and Mouron [Ergod. Th. \& Dynam. Sys. (2014) \textbf{34}, 1897--1913] constructed a map on the universal dendrite that is topologically weakly mixing but not mixing. We modify the Hoehn-Mouron example to show that there exists a transitive (even weakly mixing) dendrite map with zero topological entropy. This answers the question of Baldwin [Topology (2001) \textbf{40}, 551--569].