Automorphic measures and invariant distributions for circle dynamics

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.

arXiv comment33 pages, 3 figures. Comments are welcome

References in corpus (7)

• Rigidity of critical circle maps
• Singular measures in circle dynamics
• Rigidity of smooth critical circle maps
• Cohomological equations and invariant distributions for minimal circle diffeomorphisms
• Renormalization of bicritical circle maps
• There are no $\sigma$-finite absolutely continuous invariant measures for multicritical circle maps
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Cited by in corpus (2)

• Authomorphic measures with negative exponents for multicritical circle maps
• Renormalization of circle maps and smoothness of Arnold tongues