# Shadowing Property for the free group acting in the circle

## TL;DR

This paper investigates the shadowing property for free group actions on the circle, establishing conditions under which the property holds or fails based on the nature of the minimal set.

## Contribution

It proves that free group actions on the circle lack shadowing if the minimal set isn't a Cantor set and constructs an example where it does, depending on the minimal set.

## Key findings

- Shadowing property fails if minimal set is not a Cantor set.
- An example with a Cantor minimal set exhibits the shadowing property.
- Provides criteria linking minimal set structure to shadowing in group actions.

## Abstract

For the free group $F_2$ acting in $S^{1}$, we will prove that if the minimal set for the action is not a Cantor set, then the action does not have the shadowing property. We will also construct an example, whose minimal set is a Cantor set, that it has the shadowing property.