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 δ — 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 X n ε (or diffusion X t ε , in the continuous time case) with a small parameter ε > 0 and one is interested whether invariant measures of X n ε converge as ε → 0 to a particular invariant measure of the diffeomorphism f. To describe limiting measures I employed in [5] the Donsker-Varadhan variational formula (1) $$ {\lambda ^\varepsilon }(V) = \mathop {\sup }\limits_{\mu \in p(M)} (\int {Vd\mu } - {I^\varepsilon }(\mu )) $$ for the principal eigenvalues λ ε (V) of the operators P V ε g = P ε (e V g) where P ε is the transition operator of the Markov chain X n ε , V is a continuous function, and I(μ) 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 λε(V) converges as ε → 0 to the topological pressure Q(V + φ u ) of f corresponding to the function V + φ u where φu = -log J u (x) and J u (x) is the Jacobian of the differential Df restricted to the unstable subbundle.