On limit sets and equicontinuity in the hyperspace of continua in dimension one

TL;DR

This paper investigates the structure of limit sets and equicontinuity properties of induced maps on hyperspaces of continua, specifically in one-dimensional topological graphs, revealing conditions for almost equicontinuity and characterizing Birkhoff centers.

Contribution

It provides new insights into the dynamics of induced hyperspace maps on topological graphs, including conditions for almost equicontinuity and Birkhoff center characterization.

Findings

Induced map $ ilde{f}$ on hyperspaces of continua is almost equicontinuous on topological trees.

Characterization of the Birkhoff center for these induced maps.

Structural analysis of $ ext{omega}$-limit sets in the hyperspace context.

Abstract

The paper studies the structure of $\omega$-limit sets of map $\tilde{f}$ induced on the hyperspace $C(G)$ of all connected compact sets, by dynamical system $(G,f)$ acting on a topological graph $G$. In the case of the base space being a topological tree we additionally show that $\tilde{f}$ is always almost equicontinuous and characterize its Birkhoff center.