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Authomorphic measures with negative exponents for multicritical circle maps

Nataliya Goncharuk, Michael Yampolsky

Abstract

Authomorphic or $s$-measures for circle diffeomorphisms were introduced by R.Douady and J.-C. Yoccoz in 1999. They have multiple applications in circle dynamics, with the case $s=-1$ being particularly important for describing conjugacy classes. In arxiv:2306.13524, E. de Faria, P. Guarino and B. Nussenzveig proved existence and uniqueness of automorphic $s$-measures for multicritical circle maps for all $s>0$. The purpose of this paper is to extend these results to $s$-measures with negative values of $s$. As an application, we prove a smoothness result for irrational Arnold tongues in families of multicritical maps.

Authomorphic measures with negative exponents for multicritical circle maps

Abstract

Authomorphic or -measures for circle diffeomorphisms were introduced by R.Douady and J.-C. Yoccoz in 1999. They have multiple applications in circle dynamics, with the case being particularly important for describing conjugacy classes. In arxiv:2306.13524, E. de Faria, P. Guarino and B. Nussenzveig proved existence and uniqueness of automorphic -measures for multicritical circle maps for all . The purpose of this paper is to extend these results to -measures with negative values of . As an application, we prove a smoothness result for irrational Arnold tongues in families of multicritical maps.
Paper Structure (7 sections, 16 theorems, 51 equations)

This paper contains 7 sections, 16 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Multicritical circle maps
  3. Existence of automorphic measures
  4. $s$-measures are atomless
  5. Uniqueness of automorphic measures with negative exponents
  6. Differentiability of limits in Banach spaces
  7. (-1)-measures and smoothness of irrational Arnold tongues

Key Result

Theorem 2

Each multicritical circle map $f$ with irrational rotation number has a unique $s$-measure for any $s<0$.

Theorems & Definitions (29)

  • Definition 1
  • Theorem 2
  • Remark 3
  • Definition 4
  • Lemma 5
  • Lemma 6: Real Bounds
  • Lemma 7: Distortion lemma
  • Lemma 8
  • Theorem 9: R. Douady, J.-C. Yoccoz
  • Theorem 10
  • ...and 19 more