Arcsine law for random dynamics with a core

In their recent paper [8], G.Hata and the fourth author 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.