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.

Figures (3)

• Figure 1: The intervals $\Delta^\ast$, $\widetilde{\Delta}$ and $\widehat{\Delta}$.
• Figure 2: Relative positions of the intervals $I_n$, $I_{n+1}$, $I_n^{q_{n+1}-q_n}$, $I_n^{q_{n+1}}$, $I_{n+1}^{q_n}$, $I_{n+1}^{q_{n+1}}$, $I_n^{2 q_{n+1}}$, $I_{n+1}^{q_n+q_{n+1}}$ when $a_{n+1} \geq 2$.
• Figure 3: Relative positions of the intervals $I_n$, $I_{n+1}$, $I_n^{q_{n+1}-q_n}$, $I_n^{q_{n+1}}$, $I_{n+1}^{q_n}$, $I_{n+1}^{q_{n+1}}$, $I_n^{2 q_{n+1}}$, $I_{n+1}^{q_n+q_{n+1}}$, $I_{n+2}$, $I_{n+2}^{q_{n+1}}$, $I_{n+2}^{q_{n+1} + q_{n+2}}$ when $a_{n+1} = a_{n+2} = 1$. By applying the real bounds and Lemma \ref{['comparability of a bunch of images of atoms']}, one can see all these intervals have comparable lengths.

Theorems & Definitions (73)

• Theorem 1: Existence and uniqueness of automorphic measures
• Theorem 2: No invariant distributions
• Theorem 3
• Theorem 4: Improved Denjoy-Koksma
• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Theorem 2.4
• Lemma 2.5: Koebe distortion principle
• Remark 2.6