# Big mapping class groups and rigidity of the simple circle

## TL;DR

This paper investigates the actions of the big mapping class group of the plane minus a Cantor set on the circle, revealing a rigidity phenomenon where actions are either trivial or uniquely minimal on a special simple circle.

## Contribution

It establishes a rigidity result for the circle actions of the big mapping class group, identifying a unique minimal action on the simple circle.

## Key findings

- All actions are either trivial or semi-conjugate to the simple circle action.
- Uniqueness of the minimal simple circle action.
- Rigidity of circle actions for the big mapping class group.

## Abstract

Let $\Gamma$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\Gamma$ on the circle is either trivial or semi-conjugate to a unique minimal action on the so-called simple circle.