Limit Sets and Internal Transitivity in Free Group Actions

TL;DR

This paper extends the concept of internal chain transitivity and limit sets from classical dynamical systems to free group actions, providing characterizations in shifts of finite type and systems with shadowing.

Contribution

It introduces generalized definitions of limit sets and internal transitivity for free group actions and proves their characterizations in specific dynamical contexts.

Findings

Limit sets in free group actions are characterized by internal transitivity.

Characterizations hold in shifts of finite type.

Results apply to systems with shadowing property.

Abstract

It has been recently shown that, under appropriate hypotheses, the $\omega$-limit sets of a dynamical system are characterized by internal chain transitivity. In this paper, we examine generalizations of these ideas in the context of the action of a finitely generated free group or monoid. We give general definitions for several types of limit sets and analogous notions of internal transitivity. We then demonstrate that these limit sets are completely characterized by internal transitivity properties in shifts of finite type and general dynamical systems exhibiting a form of the shadowing property.