Exponential Law for Random Maps on Compact Manifolds

TL;DR

This paper proves that for certain random dynamical systems on manifolds, the hitting and return times follow an exponential distribution, with applications to random interval maps and parabolic maps.

Contribution

It establishes exponential laws for hitting and return times in random systems with geometric properties and decay of correlations, extending previous deterministic results.

Findings

Hitting and return times are exponentially distributed for almost every realization.

Results apply to random $C^2$ maps of the interval and parabolic maps.

Demonstrates exponential laws under geometric and decay conditions.

Abstract

We consider random dynamical systems on manifolds modeled by a skew product which have certain geometric properties and whose measures satisfy quenched decay of correlations at a sufficient rate. We prove that the limiting distribution for the hitting and return times to geometric balls are both exponential for almost every realisation. We then apply this result to random $C^2$ maps of the interval and random parabolic maps on the unit interval.