Generic chaos on dendrites

TL;DR

This paper characterizes specific dendrites where generic chaos of continuous maps can be fully described by their behavior on subdendrites with nonempty interiors, focusing on the structure of the dendrites.

Contribution

It provides a complete characterization of dendrites with this property, identifying the structural conditions related to regularity and point order.

Findings

Dendrites with all points of finite order are characterized.

Such dendrites exclude copies of the Riemann dendrite and the ω-star.

The characterization links the structure of dendrites to the behavior of chaotic maps.

Abstract

We characterize dendrites $D$ such that a continuous selfmap of $D$ is generically chaotic (in the sense of Lasota) if and only if it is generically $\varepsilon$-chaotic for some $\varepsilon>0$. In other words, we characterize dendrites on which generic chaos of a continuous map can be described in terms of the behaviour of subdendrites with nonempty interiors under iterates of the map. A dendrite $D$ belongs to this class if and only if it is completely regular, with all points of finite order (that is, if and only if $D$ contains neither a copy of the Riemann dendrite nor a copy of the $\omega$-star).