Tatiana Golenishcheva-Kutuzova, Anton Gorodetski, Victor Kleptsyn, Denis Volk, “Translation numbers define generators of <nobr>$F_k^+\to\mathrm{Homeo}_+(\mathbb S^1)$</nobr>”, Mosc. Math. J., 2014, Volume 14, Number 2,Pages <nobr>291

This article is cited in 4 papers

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

Tatiana Golenishcheva-Kutuzova^a, Anton Gorodetski^b, Victor Kleptsyn^c, Denis Volk^de

^a Moscow Center for Continuous Mathematical Education, Moscow, Russia
^b Department of Mathematics, University of California, Irvine CA 92697, USA
^c CNRS, Institute of Mathematical Research of Rennes (IRMAR, UMR 6625 du CNRS), France
^d Institute for Information Transmission Problems, Russian Academy of Sciences
^e KTH Matematik, Lindstedsvägen 25, SE-100 44 Stockholm Sweden

Abstract: We consider a minimal action of a finitely generated semigroup by homeomorphisms of the 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.

Key words and phrases: groups of homeomorphisms of the circle, rotation number, translation number, synchronization.

MSC: 37E10, 37E45, 20M20

Received: June 23, 2013; in revised form October 2, 2013

Language: English