Absolutely continuous invariant measures for random non-uniformly expanding maps
Abstract
We prove existence of (at most denumerable many) absolutely continuous invariant probability measures for random one-dimensional dynamical systems with asymptotic expansion. If the rate of expansion (Lyapunov exponents) is bounded away from zero, we obtain finitely many ergodic absolutely continuous invariant probability measures, describing the asymptotics of almost every point. We also prove a similar result for higher-dimensional random non-uniformly expanding dynamical systems. The results are consequences of the construction of such measures for skew-products with essentially arbitrary base dynamics and asymptotic expansion along the fibers. In both cases our method deals with either critical or singular points for the random maps.
Cite
@article{arxiv.1111.4540,
title = {Absolutely continuous invariant measures for random non-uniformly expanding maps},
author = {Vitor Araujo and Javier Solano},
journal= {arXiv preprint arXiv:1111.4540},
year = {2014}
}
Comments
30 pages; 2 figures. Keywords: non-uniform expansion, random dynamics, slow recurrence, singular and critical set, absolutely continuous invariant measures, skew-product. To appear in Math Z, 2014