A Variational Approach to the Random Diffeomorphisms Type Perturbations of a Hyperbolic Diffeomorphism

Let f be a diffeomorphism of a compact Riemannian manifold M. The standard model of random perturbations of f is generated by a particle which jumps from x to fx and smears with some distribution close to the $ --- function at fx. This model has the continuous time version where a flow is perturbed by a small diffusion. This approach leads to a Markov chain Xn$(or diffusion Xt$, in the continuous time case) with a small parameter $ > 0 and one is interested whether invariant measures of Xn$converge as $ textrightarrow 0 to a particular invariant measure of the diffeomorphism f. To describe limiting measures I employed in [5] the Donsker-Varadhan variational formula (1) backslashlambda ^backslashvarepsilon (V) = backslashmathop backslashsup backslashlimitsbackslashmu backslashin p(M) (backslashint Vdbackslashmu - I^backslashvarepsilon (backslashmu )) for the principal eigenvalues $$(V) of the operators PV$g = P$(eVg) where P$is the transition operator of the Markov chain Xn$, V is a continuous function, and I($) is certain lower semicontinuous convex functional on the space P(M) of probability measures on M. It turns out that if f is a hyperbolic diffeomorphism then $$(V) converges as $ textrightarrow 0 to the topological pressure Q(V + $u) of f corresponding to the function V + $uwhere $u = -log Ju(x) and Ju(x) is the Jacobian of the differential Df restricted to the unstable subbundle.