Maximal equicontinuous factor and minimal map on finitely suslinean continua

TL;DR

This paper characterizes the structure of minimal maps on finitely suslinean continua, showing they are topological circles with irrational rotations, and introduces negatively regionally proximal pairs for onto maps.

Contribution

It introduces negatively regionally proximal pairs for onto maps and proves the maximal equicontinuous factor for such maps on locally connected continua is monotone.

Findings

Maximal equicontinuous factor is monotone for onto maps on locally connected continua.

Minimal maps on finitely suslinean continua are topological circles with irrational rotations.

Provides a new characterization of minimal dynamics on certain continua.

Abstract

In this paper, we introduce the notion of negatively regionally proximal pairs of onto maps which coincides with the set of regionally proximal pair of $f^{-1}$, whenever $f$ is an homeomorphism and we prove the maximal equicontinoues factor for any onto map on a locally connected continuum is monotone. Using this, we prove that if $f$ is a minimal map on a finitely suslinean continua $X$, then $X$ must be a topological circle and $f$ some irrational rotation of circle.