Dynamical systems and random perturbations.

Abstract

This dissertation consists of two independent parts. In the first part we study the ergodic theory of surface endomorphisms. We consider non-uniformly expanding maps with generic singularities, and prove that the Pesin formula holds, which is to say that entropy is equal to the sum of the positive Lyapunov exponents if and only if the invariant probability measure in question is absolutely continuous with respect to Lebesgue measure. In the second part we study the small random perturbations of the Feigenbaum map related to the fixed point of Feigenbaum's renormalization operator for unimodal maps of the interval. We give a rigorous analysis of the changes in the geometry of the noisy attractor as noise level varies.