Moscow Mathematical Journal

Volume 14, Issue 2, April–June 2014 pp. 291–308.

Translation Numbers Define Generators of F_k⁺→Homeo₊(𝕊¹)

Authors: Tatiana Golenishcheva-Kutuzova (1), Anton Gorodetski (2), Victor Kleptsyn (3), and Denis Volk (4)
Author institution:(1) Moscow Center for Continuous Mathematical Education, Moscow, Russia
(2) Department of Mathematics, University of California, Irvine CA 92697, USA
(3) CNRS, Institute of Mathematical Research of Rennes (IRMAR, UMR 6625 du CNRS), France
(4) KTH Matematik, Lindstedsvägen 25, SE-100 44 Stockholm Sweden
Institute for Information Transmission Problems, Russian Academy of Sciences

Summary:

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.

2010 Mathematics Subject Classification. 37E10, 37E45, 20M20.

Keywords: Groups of homeomorphisms of the circle, rotation number, translation number, synchronization.

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