A.C.I.M for Random Intermittent Maps : Existence, Uniqueness and Stochastic Stability

Abstract

We study a random map $T$ which consists of intermittent maps $\{T_k{\}}_{k=1}^K$ and a position dependent probability distribution $\{p_{k,\epsilon }(x){\}}_{k=1}^K$. We prove existence of a unique absolutely continuous invariant measure (ACIM) for the random map $T$. Moreover, we show that, as $\epsilon$ 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 $T_1$. The outcome of this paper contains a first result on stochastic stability, in the strong sense, of intermittent maps.

13 pages