Equicontinuity on semi-locally connected spaces

TL;DR

This paper characterizes equicontinuous homeomorphisms on semi-locally connected compact metric spaces using a condition involving distances between iterates and subcontinua, highlighting differences from general spaces.

Contribution

It establishes a necessary and sufficient condition for equicontinuity in semi-locally connected spaces, extending understanding beyond general compact metric spaces.

Findings

Equicontinuity characterized by bounded away distances in semi-locally connected spaces.

False characterization in general compact metric spaces.

Automorphic points include points where the space is not semi-locally connected.

Abstract

We show that a homeomorphism of a semi-locally connected compact metric space is equicontinuous if and only if the distance between the iterates of a given point and a given subcontinuum (not containing that point) is bounded away from zero. This is false for general compact metric spaces. Moreover, homeomorphisms for which the conclusion of this result holds satisfy that the set of automorphic points contains those points where the space is not semi-locally connected.