Arcsine law for random dynamics with a core

TL;DR

This paper introduces a new class of random interval map dynamics called core random dynamics, demonstrating the arcsine law applies even without Markov partitions, expanding understanding of intermittent behavior in ergodic theory.

Contribution

It presents a novel framework for random interval maps that satisfy the arcsine law without relying on Markov partitions, broadening the scope of intermittent dynamics analysis.

Findings

Arcsine law holds for generalized Hata-Yano maps.

Arcsine law applies to piecewise linear Gharaei-Homburg maps.

Core random dynamics extend intermittent behavior understanding.

Abstract

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.