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. 
@article{TM_2000_228_a17,
     author = {V. Y. Kaloshin and Ya. G. Sinai},
     title = {Simple {Random} {Walks} along {Orbits} of {Anosov} {Diffeomorphisms}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {236--245},
     publisher = {mathdoc},
     volume = {228},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2000_228_a17/}
}
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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/