Law of iterated logarithm and invariance principle for one-parameter families of interval maps

TL;DR

This paper establishes the law of iterated logarithm and invariance principles for Birkhoff sums in one-parameter families of piecewise expanding interval maps, demonstrating typical statistical behavior for almost all maps.

Contribution

It proves the law of iterated logarithm and invariance principles for Birkhoff sums in generic one-parameter families of interval maps, extending previous results to broader classes.

Findings

Almost sure invariance principle for Birkhoff sums

Law of iterated logarithm holds for typical maps

Results apply to general one-parameter families

Abstract

We show that for almost every map in a transversal one-parameter family of piecewise expanding unimodal maps the Birkhoff sum of suitable observables along the forward orbit of the turning point satisfies the law of iterated logarithm. This result will follow from an almost sure invariance principle for the Birkhoff sum, as a function on the parameter space. Furthermore, we obtain a similar result for general one-parameter families of piecewise expanding maps on the interval.