Stochastic perturbations of intermittent maps

This thesis studies statistical properties of intermittent maps. We obtain three new results. First we use an Ulam-type discretization scheme to provide {\em{pointwise}} approximations for invariant densities of interval maps with a neutral fixed point. We prove that the approximate invariant density converges pointwise to the true density at a rate $C^*\cdot\frac{\ln m}{m}$, where $C^*$ is a computable fixed constant and $\frac{1}{m}$ is the mesh size of the discretization. We then study intermittent maps in a random setting. In particular, we study a random map $T$ which consists of intermittent maps $\{\tau_{k}\}_{k=1}^{K}$ and a position dependent probability distribution $\{p_{k,\varepsilon}(x)\}_{k=1}^{K}$. We prove existence of a unique absolutely continuous invariant measure (ACIM) for the random map $T$. Moreover, we show that, as $\varepsilon$ goes to zero, the invariant density of the random system $T$ converges in the $L^{1}$-norm to the invariant density of the deterministic intermittent map $\tau_{1}$. The outcome of Chapter \ref{chapACIM} contains a first result on stochastic stability, in the strong sense, of intermittent maps. Finally, we study the problem of correlation decay of random map built from finitely many intermittent maps with a common neutral fixed point. Using a Young-tower technique, we show that the map with the fastest relaxation rate dominates the asymptotics. In particular, we prove that the rate of correlation decay for the annealed dynamics of the random map is the same as the {\em sharp rate} of correlation decay for the map with the fastest relaxation rate.