Distribution of cycles for one-dimensional random dynamical systems

TL;DR

This paper studies the distribution of cycles in one-dimensional random dynamical systems generated by finitely many non-uniformly expanding Markov maps, establishing almost-sure equidistribution results and applications to random $eta$-expansions.

Contribution

It proves samplewise equidistribution of random cycles and extends Bowen's theorem to this stochastic setting, with applications to number expansions.

Findings

Almost-sure equidistribution of random cycles as periods tend to infinity

Extension of Bowen's theorem to random dynamical systems

Convergence results for random $eta$-expansions

Abstract

We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the uniqueness of equilibrium state for the associated skew product map, we establish a samplewise (quenched) almost-sure level-2 weighted equidistribution of "random cycles", with respect to a natural stationary measure as the periods of the cycles tend to infinity. This result implies an analogue of Bowen's theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms. We also prove another almost-sure convergence theorem, as well as an averaged (annealed) theorem that is related to semigroup actions. We apply our results to the random $\beta$-expansion of real numbers, and obtain almost-sure convergences of average digital quantities in random $\beta$-expansions of random cycles that do not follow from the application of the ergodic theorems of Birkhoff or Kakutani. Our main results are applicable to random dynamical systems generated by finitely many maps with common neutral fixed points.