Symmetric random walks on Homeo+(R)

• B. Deroin ^[1] ; V. Kleptsyn ^[4] ; A. Navas ^[2] ; K. Parwani ^[3]
  1. [1] University of Paris-Sud

     University of Paris-Sud

     Arrondissement de Palaiseau, Francia

     
  2. [2] Universidad de Santiago de Chile

     Universidad de Santiago de Chile

     Santiago, Chile

     
  3. [3] Eastern Illinois University

     Eastern Illinois University

     Township of Charleston, Estados Unidos

     
  4. [4] Institut de Recherches Mathématiques de Rennes

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 3, 2, 2013, págs. 2066-2089
• Idioma: inglés
• DOI: 10.1214/12-AOP784
• Texto completo no disponible (Saber más ...)
• Resumen

  • We study symmetric random walks on finitely generated groups of orientation-preserving homeomorphisms of the real line. We establish an oscillation property for the induced Markov chain on the line that implies a weak form of recurrence. Except for a few special cases, which can be treated separately, we prove a property of “global stability at a finite distance”: roughly speaking, there exists a compact interval such that any two trajectories get closer and closer whenever one of them returns to the compact interval. The probabilistic techniques employed here lead to interesting results for the study of group actions on the line. For instance, we show that under a suitable change of the coordinates, the drift of every point becomes zero provided that the action is minimal. As a byproduct, we recover the fact that every finitely generated group of homeomorphisms of the real line is topologically conjugate to a group of (globally) Lipschitz homeomorphisms. Moreover, we show that such a conjugacy may be chosen in such a way that the displacement of each element is uniformly bounded.