Translation numbers define generators of $F_k^+\to {\text{\rm Homeo}_+}(\mathbb{S}^1)$

TL;DR

This paper shows that for a minimal semigroup action on a circle, the translation numbers of generators uniquely determine the generators up to a continuous change of coordinates, using synchronization properties of circle homeomorphisms.

Contribution

It introduces a method to recover generators of a semigroup action on the circle from translation numbers, leveraging synchronization properties and Antonov's theorem.

Findings

Translation numbers uniquely determine generators up to conjugacy.

Synchronization properties are key in the proof.

The approach applies to minimal actions of finitely generated semigroups.

Abstract

We consider a minimal action of a finitely generated semigroup by homeomorphisms of a circle, and show that the collection of translation numbers of individual elements completely determines the set of generators (up to a common continuous change of coordinates). One of the main tools used in the proof is the synchronization properties of random dynamics of circle homeomorphisms: Antonov's theorem and its corollaries.