Sensitive open map semigroups on Peano continua having a free arc

TL;DR

This paper characterizes Peano continua with a free arc that admit sensitive, commutative, open map semigroups, showing such spaces are either arcs or circles, thus linking topological structure with dynamical sensitivity.

Contribution

It proves that Peano continua with a free arc supporting a sensitive commutative semigroup of open maps are necessarily arcs or circles, revealing a topological-dynamical classification.

Findings

Spaces are either arcs or circles.

Sensitive semigroups imply strong topological restrictions.

Open maps preserve the continuum's structure.

Abstract

Let $X$ be a Peano continuum having a free arc and let $C^0(X)$ be the semigroup of continuous self-maps of $X$. A subsemigroup $F\subset C^0(X)$ is said to be sensitive, if there is some constant $c>0$ such that for any nonempty open set $U\subset X$, there is some $f\in F$ such that the diameter ${\rm diam}(f(U))>c$. We show that if $X$ admits a sensitive commutative subsemigroup $F$ of $C^0(X)$ consisting of continuous open maps, then either $X$ is an arc, or $X$ is a circle.