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

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.

Central Limit Theorem for the Modulus of Continuity of Averages of Observables on Transversal Families of Piecewise Expanding Unimodal Maps

“…From this it follows that J > 0 and that J(f t , v t ) does not changes signs for t ∈ [a, b]. In [19], Schnellmann proved that t → σ t is Hölder continuous.…”

“…Conditions (I), (II) and (III) are exactly those that appears in Schnellmann [19], with obvious cosmetic modifications. …”

“…We quote almost verbatim the definition of the Banach spaces V α given in [19]. The spaces V α were introduced by Keller [10].…”

“…Here, we introduce the notation of partitions following Schnellmann in [19]. Let us denote by K(t) = [f 2 t (c), f t (c)] the support of ρ t .…”

“…We quote almost verbatim the definition of the almost sure invariant principle given in [19]. Definition 3.2.…”