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

TL;DR

This paper proves the existence and uniqueness of an absolutely continuous invariant measure for a class of random intermittent maps and demonstrates their stochastic stability as randomness diminishes.

Contribution

It establishes the first strong stochastic stability result for intermittent maps, showing convergence of invariant densities under random perturbations.

Findings

Existence of a unique ACIM for the random map.

Convergence of invariant density to the deterministic case as randomness vanishes.

First strong stochastic stability result for intermittent maps.

Abstract

We study a random map $T$ which consists of intermittent maps $\{T_{k}\}_{k=1}^{K}$ and a position dependent probability distribution $\{p_{k,\varepsilon}(x)\}_{k=1}^{K}$. 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 $T_{1}$. The outcome of this paper contains a first result on stochastic stability, in the strong sense, of intermittent maps.