Arcsine and Darling--Kac laws for piecewise linear random interval maps

Abstract

We give an example of a piecewise linear random interval map satisfying arcsine and Darling-Kac laws, which are analogous to Thaler's arcsine and Aaronson's Darling-Kac laws for the Boole transform. It is constructed by random switch of two piecewise linear maps with attracting fixed points, which behave as if they were indifferent fixed points of a deterministic map.

Arcsine law for random dynamics with a core

“…From the standard argument of simple symmetric random walks on Z (cf. [8]), for 0 < s < 1 we have as N tends to infinity. Now we can apply Theorem 2.3 to the specific τ, X, m, A, A ± given in (4.9).…”

“…Although the existence of indifferent fixed points of maps in the above works is indispensable for the generalized arcsine law of intermittent dynamics, G. Hata and the fourth author recently showed in [8] that random iterations of two piecewise linear interval maps without indifferent periodic points, (HY)…”

“…The key in the proof of the arcsine law in [8] 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 dynamics.…”

“…

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.

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