Sensitive group actions on regular curves of almost $\leq n$ order

TL;DR

This paper investigates the dynamics of group actions on regular curves with bounded boundary complexity, showing that sensitive actions imply the presence of free subsemigroups and positive entropy, and excluding nilpotent group actions.

Contribution

It establishes a connection between sensitivity, boundary conditions, and algebraic properties of acting groups on regular curves.

Findings

Sensitive actions imply free subsemigroups in the acting group.

Such actions have positive geometric entropy.

Nilpotent group actions cannot be sensitive on these curves.

Abstract

Let $X$ be a regular curve and $n$ be a positive integer such that for every nonempty open set $U\subset X$, there is a nonempty connected open set $V\subset U$ with the cardinality $|\partial_X(V)|\leq n$. We show that if $X$ admits a sensitive action of a group $G$, then $G$ contains a free subsemigroup and the action has positive geometric entropy. As a corollary, $X$ admits no sensitive nilpotent group action.