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

Tatiana Golenishcheva-Kutuzova (1) , Anton Gorodetski (2) , Victor A. Kleptsyn (3) , Denis Volk (4)

1 MCCME - Moscow Center for Continuous Mathematical Education
2 Department of mathematics
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 KTH Matematik

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.

Mots clés: Groups of homeomorphisms of the circle; rotation number; translation number; synchronization

Domaines: Systèmes dynamiques [math.DS]; Théorie des groupes [math.GR]

Tatiana Golenishcheva-Kutuzova, Anton Gorodetski, Victor A. Kleptsyn, Denis Volk. Translation numbers define generators of $F_k^+\to {\text{\rm Homeo}_+}(\mathbb{S}^1)$. Moscow Mathematical Journal, 2014, 14 (2), pp.291-308. ⟨hal-00954935⟩