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

@inproceedings{Hata2021ArcsineAD, title={Arcsine and Darling--Kac laws for piecewise linear random interval maps}, author={Genji Hata and Kouji Yano}, year={2021} }

We give examples of piecewise linear random interval maps satisfying arcsine and Darling–Kac laws, which are analogous to Thaler’s arcsine and Aaronson’s Darling– Kac laws for the Boole transform. They are constructed by random switch of two piecewise linear maps with attracting or repelling fixed points, which behave as if they were indifferent fixed points of a deterministic map.

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SHOWING 1-10 OF 40 REFERENCES

Functional limit theorem for occupation time processes of intermittent maps

- Mathematics
- 2018

We establish a functional limit theorem for the joint-law of occupations near and away from indifferent fixed points of interval maps, and of waits for the occupations away from these points, in the… Expand

Multiray generalization of the arcsine laws for occupation times of infinite ergodic transformations

- Mathematics
- Transactions of the American Mathematical Society
- 2019

We prove that the joint distribution of the occupation time ratios for ergodic transformations preserving an infinite measure converges in the sense of strong distribution convergence to Lamperti's… Expand

A limit theorem for sojourns near indifferent fixed points of one-dimensional maps

- Mathematics
- Ergodic Theory and Dynamical Systems
- 2002

For a class of one-dimensional maps with two indifferent fixed points of the same order, a parallel to the arc-sine law for the number of positive partial sums of a real random walk is proved.

Distributional limit theorems in infinite ergodic theory

- Mathematics
- 2006

We present a unified approach to the Darling-Kac theorem and the arcsine laws for occupation times and waiting times for ergodic transformations preserving an infinite measure. Our method is based on… Expand

Statistics of close visits to the indifferent fixed point of an interval map

- Mathematics
- 1993

We study a dynamical system defined by a map of the interval [0, 1] which has 0 as an indifferent fixed point but is otherwise expanding. We prove that the sequence of successive entrance times in a… Expand

Decay of correlation for random intermittent maps

- Mathematics
- 2014

We study a class of random transformations built over finitely many intermittent maps sharing a common indifferent fixed point. Using a Young-tower technique, we show that the map with the fastest… Expand

Mixing rates and limit theorems for random intermittent maps

- Mathematics
- 2015

We study random transformations built from intermittent maps on the unit interval that share a common neutral fixed point. We focus mainly on random selections of Pomeu-Manneville-type maps… Expand

Decay of correlations for piecewise smooth maps with indifferent fixed points

- Mathematics
- Ergodic Theory and Dynamical Systems
- 2004

We consider a piecewise smooth expanding map f on the unit interval that has the form $f(x)=x+x^{1+\gamma}+o(x^{1+\gamma})$ near 0, where $0<\gamma < 1$. We prove by showing both lower and upper… Expand

Statistical properties of long return times in type I intermittency

- Mathematics
- 1995

We study a class of maps of the unit interval with a neutral fixed point such äs those modelling Pomeau-Manneville type l intermittency. We construct the invariant ergodic probability measure… Expand

Infinite measure preserving transformations with compact first regeneration

- Mathematics
- 2007

We study ergodic infinite measure preserving transformations T possessing reference sets of finite measure for which the set of densities of the conditional distributions given a first return (or… Expand