Linear response for one dimensional systems: formulae and rigorous numerics

This thesis studies the approximation of the long time statistical behaviour of one dimensional discrete time dynamical systems. First we find response formulae for a wide class of dynamical systems and apply them to the Gauss-Renyi random map to approximate the invariant density for systems near the deterministic Gauss map. We use these estimates to approximate the frequencies each digit appears in random continued fraction expansions of numbers in [−1, 1].

We then give a method for approximating the invariant densities and linear responses for a class of intermittent maps using induced maps, and go on approximate the linear response and invariant density of Pomeau-Manneville maps and gain rigorous error bounds in L¹.