Automorphic measures and invariant distributions for circle dynamics

TL;DR

This paper extends the existence and uniqueness of automorphic measures to multicritical circle maps and explores their implications for invariant distributions and inequalities in circle dynamics.

Contribution

It proves the existence and uniqueness of automorphic measures for multicritical circle maps, generalizing previous results for smooth circle diffeomorphisms.

Findings

Invariant distributions of order 1 are one-dimensional, spanned by the invariant measure.

Provides an improved Denjoy-Koksma inequality for multicritical circle maps.

Establishes the uniqueness of automorphic measures for a broader class of circle maps.

Abstract

Let $f$ be a $C^{1+bv}$ circle diffeomorphism with irrational rotation number. As established by Douady and Yoccoz in the eighties, for any given $s>0$ there exists a unique automorphic measure of exponent $s$ for $f$. In the present paper we prove that the same holds for multicritical circle maps, and we provide two applications of this result. The first one, is to prove that the space of invariant distributions of order 1 of any given multicritical circle map is one-dimensional, spanned by the unique invariant measure. The second one, is an improvement over the Denjoy-Koksma inequality for multicritical circle maps and absolutely continuous observables.