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 Mosc. Math. J., 2014, Volume 14, Number 2, Pages 339–365 (Mi mmj525)

This article is cited in 13 scientific papers (total in 13 papers)

Physical measures for nonlinear random walks on interval

V. Kleptsyna, D. Volkbc

a CNRS, Institut de Recherche Mathematique de Rennes (IRMAR, UMR 6625 CNRS)
b University of Rome "Tor Vergata"
c Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: 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 $t\to+\infty$, and to the repeller as $t\to-\infty$. The attractors support ergodic hyperbolic physical measures.

Key words and phrases: random walks, stationary measures, dynamical systems, attractors, partial hyperbolicity, skew products.

DOI: https://doi.org/10.17323/1609-4514-2014-14-2-339-365

Full text: http://www.mathjournals.org/.../2014-014-002-008.html
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Bibliographic databases:

MSC: Primary 82B41, 82C41, 60G50; Secondary 37C05, 37C20, 37C70, 37D45
Received: July 7, 2013
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Citation: V. Kleptsyn, D. Volk, “Physical measures for nonlinear random walks on interval”, Mosc. Math. J., 14:2 (2014), 339–365

Citation in format AMSBIB
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\yr 2014
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\pages 339--365
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. K. Shinohara, “On the minimality of semigroup actions on the interval which are $C^1$-close to the identity”, Proc. London Math. Soc., 109:5 (2014), 1175–1202
2. V. Kleptsyn, D. Volk, “Nonwandering sets of interval skew products”, Nonlinearity, 27:7 (2014), 1595–1601
3. M. Gharaei, A. J. Homburg, “Skew products of interval maps over subshifts”, J. Difference Equ. Appl., 22:7 (2016), 941–958
4. A. Okunev, “Milnor attractors of skew products with the fiber a circle”, J. Dyn. Control Syst., 23:2 (2017), 421–433
5. M. Gharaei, A. J. Homburg, “Random interval diffeomorphisms”, Discrete Contin. Dyn. Syst. Ser. S, 10:2 (2017), 241–272
6. J. de Simoi, C. Liverani, Ch. Poquet, D. Volk, “Fast-slow partially hyperbolic systems versus Freidlin–Wentzell random systems”, J. Stat. Phys., 166:3-4 (2017), 650–679
7. A. V. Okunev, I. S. Shilin, “On the attractors of step skew products over the Bernoulli shift”, Proc. Steklov Inst. Math., 297 (2017), 235–253
8. L. J. Diaz, K. Gelfert, M. Rams, “Nonhyperbolic step skew-products: ergodic approximation”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 34:6 (2017), 1561–1598
9. Yu. Ilyashenko, I. Shilin, “Attractors and skew products”, Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, eds. A. Katok, Y. Pesin, F. Hertz, Amer. Math. Soc., 2017, 155–175
10. A. J. Homburg, “Synchronization in minimal iterated function systems on compact manifolds”, Bull. Braz. Math. Soc., 49:3 (2018), 615–635
11. C. P. Walkden, T. Withers, “Invariant graphs of a family of non-uniformly expanding skew products over Markov maps”, Nonlinearity, 31:6 (2018), 2726–2755
12. M. Zaj, A. Fakhari, F. H. Ghane, A. Ehsani, “Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus”, Discret. Contin. Dyn. Syst., 38:4 (2018), 1777–1807
13. L. J. Diaz, E. Matias, “Stability of the Markov operator and synchronization of Markovian random products”, Nonlinearity, 31:4 (2018), 1782–1806
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