# 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

**Summary: **

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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