Simple Random Walks along Orbits of Anosov Diffeomorphisms

and non-symmetric otherwise. We implicitly assume that both integrals ∫ ln p(x) dμ(x) and ∫ ln(1 − p(x)) dμ(x) are finite. It is natural to raise the following questions: A) Does simple random walk have an invariant measure absolutely continuous wrt μ. Denote by ξx(t) ∈ {Smx,−∞ < m < ∞} the position at time t of the moving particle which starts at x. B) What is the limiting distribution of ξx(n) on the space M as n →∞. If ξx(t) = S xx, then kx(t+1)−kx(t) = ±1, i.e. kx(t) is a trajectory of a simple random walk, kx(0) = 0. The transition probabilities of this walk are determited by the dynamics of S and the random media function p. The density π (wrt μ) of an invariant measure for our Markov chain satisfies the equation