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$C^1$ Robust Rigidity for Bi-critical Circle Maps

Gabriela Estevez

TL;DR

This work proves a $C^1$ rigidity result for bi-critical circle maps: if two $C^3$ bi-critical maps share the same signature and their renormalizations around corresponding critical points converge exponentially in the $C^2$ topology, then they are $C^1$-conjugate. The authors develop the two-bridges partition to control tangencies without imposing rotation-number restrictions, and they combine real bounds, Schwarzian/Koeffe distortion, and tubular-coordinate techniques to propagate precise geometric estimates across scales. A central contribution is establishing a three-condition criterion (via refined partitions and endpoint/interval controls) that guarantees a $C^1$ conjugacy, even for unbounded rotation numbers. The results bridge prior $C^{1+eta}$ rigidity under measure-theoretic rotation-number conditions with a robust $C^1$ outcome applicable to arbitrary irrational rotations. This advances smooth rigidity theory for multi-critical dynamics and clarifies how renormalization-driven geometry governs conjugacy regularity.

Abstract

We prove that two topologically conjugate bi-critical circle maps whose signatures are the same, and whose renormalizations converge together exponentially fast in the $C^2$-topology, are $C^1$ conjugate.

$C^1$ Robust Rigidity for Bi-critical Circle Maps

TL;DR

This work proves a rigidity result for bi-critical circle maps: if two bi-critical maps share the same signature and their renormalizations around corresponding critical points converge exponentially in the topology, then they are -conjugate. The authors develop the two-bridges partition to control tangencies without imposing rotation-number restrictions, and they combine real bounds, Schwarzian/Koeffe distortion, and tubular-coordinate techniques to propagate precise geometric estimates across scales. A central contribution is establishing a three-condition criterion (via refined partitions and endpoint/interval controls) that guarantees a conjugacy, even for unbounded rotation numbers. The results bridge prior rigidity under measure-theoretic rotation-number conditions with a robust outcome applicable to arbitrary irrational rotations. This advances smooth rigidity theory for multi-critical dynamics and clarifies how renormalization-driven geometry governs conjugacy regularity.

Abstract

We prove that two topologically conjugate bi-critical circle maps whose signatures are the same, and whose renormalizations converge together exponentially fast in the -topology, are conjugate.
Paper Structure (17 sections, 22 theorems, 124 equations, 3 figures)

This paper contains 17 sections, 22 theorems, 124 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multicritical circle maps
  4. Combinatorics of multicritical circle maps
  5. Renormalization of bi-critical circle maps
  6. Real bounds
  7. Two tools: Schwarzian derivative and Koebe Distortion Principle
  8. Controlled bi-critical commuting pairs
  9. The two-bridges partition $\widehat{\mathcal{P}}_n$
  10. Properties of the two-bridges partition
  11. Idea of the proof
  12. Un-bounded rotation number
  13. Tubular coordinates
  14. Key estimates for points in $\widehat{E}_{n+1}$
  15. Key estimates for intervals of $\mathcal{\widehat{P}}_{n+1}$
  16. ...and 2 more sections

Key Result

Theorem A

Let $f$ and $g$ be two $C^3$ bi-critical circle maps with the same signature. If the renormalizations of $f$ and $g$ around corresponding critical points converge together exponentially fast in the $C^2$ topology, then $f$ and $g$ are $C^1$-conjugated.

Figures (3)

  • Figure 1: The $n-$th renormalization (in black) and the $(n+1)$-th pre-renormalization (in blue) of a bi-critical circle map $f$ around the critical point $c_0$.
  • Figure 2: The two-bridges partition $\widehat{\mathcal{P}}_{n+1}|_{I_n(c_0)}$ in Case 1.
  • Figure 3: In this figure $r(n)>L$ and then the center $z_{n}$ of $M_{n,L}^f$ is defined

Theorems & Definitions (44)

  • Theorem A
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.1: Universal Real Bounds
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Definition 2.3
  • Theorem 2.6
  • ...and 34 more