Distribution of cycles for one-dimensional random dynamical systems

Shintaro Suzuki, Hiroki Takahasi

Published 2021-08-12Version 1

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 the equilibrium state of product form, we establish an almost-sure weighted equidistribution of cycles with respect to a natural stationary measure, as the "periods" of the cycles tend to infinity. This result is an analogue of Bowen's theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms in random setup. We also prove averaging results over all samples, as well as another samplewise result. We apply our result to the random $\beta$-expansion of real numbers, and obtain a new formula for the mean relative frequencies of digits in the series expansion.