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Moscow Mathematical Journal

Volume 14, Issue 2, April–June 2014  pp. 339–365.

Physical Measures for Nonlinear Random Walks on Interval

Authors:  V. Kleptsyn (1) and D. Volk (2)
Author institution:(1) CNRS, Institut de Recherche Mathematique de Rennes (IRMAR, UMR 6625 CNRS)
(2) University of Rome “Tor Vergata” and
Institute for Information Transmission Problems, Russian Academy of Sciences


A one-dimensional confined nonlinear random walk is a tuple of N diffeomorphisms of the unit interval driven by a probabilistic Markov chain. For generic such walks, we obtain a geometric characterization of their ergodic stationary measures and prove that all of them have negative Lyapunov exponents. These measures appear to be probabilistic manifestations of physical measures for certain deterministic dynamical systems. These systems are step skew products over transitive subshifts of finite type (topological Markov chains) with the unit interval fiber.

For such skew products, we show there exist only finite collection of alternating attractors and repellers; we also give a sharp upper bound for their number. Each of them is a graph of a continuous map from the base to the fiber defined almost everywhere w.r.t. any ergodic Markov measure in the base. The orbits starting between the adjacent attractor and repeller tend to the attractor as t → +∞, and to the repeller as t → −∞. The attractors support ergodic hyperbolic physical measures.

2010 Mathematics Subject Classification. Primary: 82B41, 82C41, 60G50; Secondary: 37C05, 37C20, 37C70, 37D45.

Keywords: Random walks, stationary measures, dynamical systems, attractors, partial hyperbolicity, skew products.

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