On the entropy for group actions on the circle

TL;DR

This paper proves that for finitely generated groups acting by $C^2$ diffeomorphisms on the circle, the entropy of the entire action matches the entropy on the non-wandering set, linking global and local dynamical complexity.

Contribution

It establishes a precise equality between the entropy of the group action and its restriction to the non-wandering set for $C^2$ circle diffeomorphisms.

Findings

Entropy of the group action equals entropy on the non-wandering set.

Results apply to finitely generated groups of $C^2$ circle diffeomorphisms.

Provides a link between global and local dynamical complexity.

Abstract

We show that for a finitely generated group of $C^2$ circle diffeomorphisms, the entropy of the action equals the entropy of the restriction of the action to the non-wandering set.