Arcsine law for random dynamics with a core

Abstract

In their recent paper, Hata and Yano (2021 Stoch. Dyn. 2350006) first gave an example of random iterations of two piecewise linear interval maps without (deterministic) indifferent periodic points for which the arcsine law—a characterization of intermittent dynamics in infinite ergodic theory—holds. The key in the proof of the result is the existence of a Markov partition preserved by each interval maps. In the present paper, we give a class of random iterations of two interval maps without indifferent periodic points but satisfying the arcsine law, by introducing a concept of core random dynamics. As applications, we show that the generalized arcsine law holds for generalized Hata–Yano maps and piecewise linear versions of Gharaei–Homburg maps, both of which do not have a Markov partition in general.