Geodesic
Simple Random Walks along Orbits of Anosov Diffeomorphisms
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Problems of the modern mathematical physics, Tome 228 (2000), pp. 236-245.

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We consider a Markov chain whose phase space is a $d$-dimensional torus. A point $x$ jumps to $x+\omega$ with probability $p(x)$ and to $x-\omega$ with probability $1-p(x)$. For Diophantine $\omega$ and smooth $p$ we prove that this Maslov chain has an absolutely continuous invariant measure and the distribution of any point after $n$ steps converges to this measure. 
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     author = {V. Y. Kaloshin and Ya. G. Sinai},
     title = {Simple {Random} {Walks} along {Orbits} of {Anosov} {Diffeomorphisms}},
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}
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V. Y. Kaloshin; Ya. G. Sinai. Simple Random Walks along Orbits of Anosov Diffeomorphisms. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Problems of the modern mathematical physics, Tome 228 (2000), pp. 236-245. http://geodesic.mathdoc.fr/item/TM_2000_228_a17/

[1] Bowen R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes Math., 470, Springer-Verl., Berlin–New York, 1975 | MR | Zbl

[2] Billingsley P., Convergence of probability measures, Wiley, N. Y., 1968 | MR | Zbl

[3] Golosov A., “Localization of random walks in one-dimensional random environment”, Commun. Math. Phys., 92 (1984), 491–506 | DOI | MR | Zbl

[4] Feller W., Introduction to probability theory and its application, v. 2, Wiley, New York–London, 1966 | MR | Zbl

[5] Kesten H., “The limit distribution of Sinai's random walk in a random environment”, Physica A, 138 (1986), 299–306 | DOI | MR

[6] Kaloshin V., Sinai Ya., “Nonsymmetric random walks along orbits of ergodic automorphisms”, On Dobrushin's way. From probability theory to statistical physics, Amer. Math. Soc. Transl., 198, Amer. Math. Soc., Providence, R.I., 2000, 109–115 | MR | Zbl

[7] Sinai Ya. G., “Predelnoe povedenie odnomernogo sluchainogo bluzhdaniya v sluchainoi srede”, Teoriya veryatn. i ee prim., 27 (1982), 247–258 | MR

[8] Sinai Ya., “Simple random walks on tori”, J. Stat. Phys., 94:3–4 (1999), 695–708 | DOI | MR | Zbl

[9] Sinai Ya., Topics in ergodic theory, Princeton Math. Ser., 44 \end {thebibliography}, Princeton Univ. Press, Princeton, N.J., 1994 | MR | Zbl