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June  2021, 41(6): 2891-2905. doi: 10.3934/dcds.2020390

A symmetric Random Walk defined by the time-one map of a geodesic flow

Pablo D. Carrasco , and  Túlio Vales

Av. Presidente Antônio Carlos 6627, Belo Horizonte-MG, BR31270-901

* Corresponding author: Pablo D. Carrasco

Received  August 2020 Revised  October 2020 Published  December 2020

In this note we consider a symmetric Random Walk defined by a $ (f, f^{-1}) $ Kalikow type system, where $ f $ is the time-one map of the geodesic flow corresponding to an hyperbolic manifold. We provide necessary and sufficient conditions for the existence of an stationary measure for the walk that is equivalent to the volume in the corresponding unit tangent bundle. Some dynamical consequences for the Random Walk are deduced in these cases.

Keywords: Random walk in a random environment, stationary measures, geodesic flow.
Mathematics Subject Classification: Primary: 537C40, 37D30; Secondary: 60K37, 37H99.
Citation: Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390
References:
[1]

D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209 pp.  Google Scholar

[2]

A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows, Journal of the European Mathematical Society, 17 (2015), 1435-1462.  doi: 10.4171/JEMS/534.  Google Scholar

[3]

A. Candel and L. Conlon, Foliations I, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/gsm/023.  Google Scholar

[4]

J.-P. Conze and Y. Guivarc'h, Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique, Colloquium Mathematicum, 84 (2000), 457-480.  doi: 10.4064/cm-84/85-2-457-480.  Google Scholar

[5]

D. Dolgopyat, B. Fayad, and M. Saprykina, Erratic behavior for 1-dimensional Random Walks in a Liouville quasi-periodic environment., preprint, 2019, arXiv: 1901.10709. Google Scholar

[6]

M. Gorodin and B. Lifsic, Central limit theorem for stationary Markov processes, In Third Vilnius Conference on Probability and Statistics, volume 1, 1981,147–148. Google Scholar

[7] B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University Press, New York, 2003.  doi: 10.1017/CBO9780511998188.  Google Scholar
[8]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A Survey of Partially Hyperbolic Dynamics, In Partially Hyperbolic Dynamics, Laminations and Teichmüller Flow, Fields Institute Communications, vol. 51, 2007, 35–88.  Google Scholar

[9]

M. W. Hirsch, C. C. Pugh, and M. I. Shub, Invariant Manifolds, Springer Berlin Heidelberg, 1977. Google Scholar

[10]

V. Kaloshin and Y. Sinai, Simple random walks along orbits of Anosov diffeomorphisms, Tr. Mat. Inst. Steklova, 228 (2000), 236-245.   Google Scholar

[11]

A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Mathematical Research Letters, 3 (1996), 191-210.  doi: 10.4310/MRL.1996.v3.n2.a6.  Google Scholar

[12]

Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.  Google Scholar

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C. Kipnis and S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Communications Math. Physics, 104 (1986), 1-19.  doi: 10.1007/BF01210789.  Google Scholar

[14]

J. Neveu and A. Feinstein, Mathematical Foundations of the Calculus of Probability, Holden-Day, Inc. San Francisco, Calif.-London-Amsterdam, 1965.  Google Scholar

[15]

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society, Zürich, 2004. doi: 10.4171/003.  Google Scholar

[16]

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS), 2 (2000), 1-52.  doi: 10.1007/s100970050013.  Google Scholar

[17]

F. Rodriguez-Hertz, J. Rodriguez-Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Inventiones Mathematicae, 172 (2008), 353-381.  doi: 10.1007/s00222-007-0100-z.  Google Scholar

[18]

F. Rodriguez-Hertz, J. Rodriguez-Hertz and R. Ures, A non-dynamically coherent example on $\mathbb{T}^3$, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 33 (2016), 1023-1032.  doi: 10.1016/j.anihpc.2015.03.003.  Google Scholar

[19]

V. Rokhlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk., 22 (1967), 3-56.   Google Scholar

[20]

Y. Sinai, Simple random walks on tori, Journal of Statistical Physics, 94 (1999), 695-708.  doi: 10.1023/A:1004564824697.  Google Scholar

[21]

W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory and Dynamical Systems, 6 (1986), 449-473.  doi: 10.1017/S0143385700003606.  Google Scholar

[22]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75–165.  Google Scholar

[23]

O. Zeitouni, Random walks in random environment, In Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004,189–312. doi: 10.1007/978-3-540-39874-5_2.  Google Scholar

show all references

References:
[1]

D. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209 pp.  Google Scholar

[2]

A. Avila, M. Viana and A. Wilkinson, Absolute continuity, Lyapunov exponents and rigidity I: Geodesic flows, Journal of the European Mathematical Society, 17 (2015), 1435-1462.  doi: 10.4171/JEMS/534.  Google Scholar

[3]

A. Candel and L. Conlon, Foliations I, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/gsm/023.  Google Scholar

[4]

J.-P. Conze and Y. Guivarc'h, Marches en milieu aléatoire et mesures quasi-invariantes pour un système dynamique, Colloquium Mathematicum, 84 (2000), 457-480.  doi: 10.4064/cm-84/85-2-457-480.  Google Scholar

[5]

D. Dolgopyat, B. Fayad, and M. Saprykina, Erratic behavior for 1-dimensional Random Walks in a Liouville quasi-periodic environment., preprint, 2019, arXiv: 1901.10709. Google Scholar

[6]

M. Gorodin and B. Lifsic, Central limit theorem for stationary Markov processes, In Third Vilnius Conference on Probability and Statistics, volume 1, 1981,147–148. Google Scholar

[7] B. Hasselblatt and A. Katok, A First Course in Dynamics, Cambridge University Press, New York, 2003.  doi: 10.1017/CBO9780511998188.  Google Scholar
[8]

F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, A Survey of Partially Hyperbolic Dynamics, In Partially Hyperbolic Dynamics, Laminations and Teichmüller Flow, Fields Institute Communications, vol. 51, 2007, 35–88.  Google Scholar

[9]

M. W. Hirsch, C. C. Pugh, and M. I. Shub, Invariant Manifolds, Springer Berlin Heidelberg, 1977. Google Scholar

[10]

V. Kaloshin and Y. Sinai, Simple random walks along orbits of Anosov diffeomorphisms, Tr. Mat. Inst. Steklova, 228 (2000), 236-245.   Google Scholar

[11]

A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Mathematical Research Letters, 3 (1996), 191-210.  doi: 10.4310/MRL.1996.v3.n2.a6.  Google Scholar

[12]

Y. Kifer, Ergodic Theory of Random Transformations, Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.  Google Scholar

[13]

C. Kipnis and S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Communications Math. Physics, 104 (1986), 1-19.  doi: 10.1007/BF01210789.  Google Scholar

[14]

J. Neveu and A. Feinstein, Mathematical Foundations of the Calculus of Probability, Holden-Day, Inc. San Francisco, Calif.-London-Amsterdam, 1965.  Google Scholar

[15]

Y. Pesin, Lectures on Partial Hyperbolicity and Stable Ergodicity, European Mathematical Society, Zürich, 2004. doi: 10.4171/003.  Google Scholar

[16]

C. Pugh and M. Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS), 2 (2000), 1-52.  doi: 10.1007/s100970050013.  Google Scholar

[17]

F. Rodriguez-Hertz, J. Rodriguez-Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Inventiones Mathematicae, 172 (2008), 353-381.  doi: 10.1007/s00222-007-0100-z.  Google Scholar

[18]

F. Rodriguez-Hertz, J. Rodriguez-Hertz and R. Ures, A non-dynamically coherent example on $\mathbb{T}^3$, Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 33 (2016), 1023-1032.  doi: 10.1016/j.anihpc.2015.03.003.  Google Scholar

[19]

V. Rokhlin, Lectures on the entropy theory of transformations with invariant measure, Uspehi Mat. Nauk., 22 (1967), 3-56.   Google Scholar

[20]

Y. Sinai, Simple random walks on tori, Journal of Statistical Physics, 94 (1999), 695-708.  doi: 10.1023/A:1004564824697.  Google Scholar

[21]

W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory and Dynamical Systems, 6 (1986), 449-473.  doi: 10.1017/S0143385700003606.  Google Scholar

[22]

A. Wilkinson, The cohomological equation for partially hyperbolic diffeomorphisms, Astérisque, 358 (2013), 75–165.  Google Scholar

[23]

O. Zeitouni, Random walks in random environment, In Lecture Notes in Math., vol. 1837, Springer, Berlin, 2004,189–312. doi: 10.1007/978-3-540-39874-5_2.  Google Scholar

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