On the dynamics of subcontinua of a tree

TL;DR

This paper investigates the long-term behavior of subcontinua in tree maps, showing they tend toward periodicity or degeneration, and links entropy properties of the system to its functional envelope.

Contribution

It establishes a dichotomy for subcontinua dynamics in tree maps and connects entropy of the system with that of its functional envelope.

Findings

Subcontinua are either asymptotically periodic or degenerate.

Zero entropy of the system implies zero entropy of the functional envelope.

Provides a characterization of long-term behavior in tree map dynamics.

Abstract

Given a tree map $f:T\to T$, we study the dynamics of subcontinua of $T$ under action of $f$. In particular, we prove that a subcontinuum of $T$ is either asymptotically periodic or asymptotically degenerate. As an application of this result, we show that zero topological entropy of the system $(T,f)$ implies zero topological entropy of its functional envelope (endowed with the Hausdorff metric).