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Topology of irrationally indifferent attractors

Davoud Cheraghi

TL;DR

This work classifications the local attractors of holomorphic maps with irrationally indifferent fixed points, proving a trichotomy for the post-critical set based on the arithmetic of the rotation number $\alpha$: a Jordan curve (Herman type), a one-sided hairy Jordan curve (Brjuno but not Herman), or a Cantor bouquet (non-Brjuno). Central to the analysis is a toy topological model that captures the essence of near-parabolic renormalisation and the role of arithmetic types, together with a thorough comparison between the toy model and the analytic renormalisation via explicit coordinate changes and contraction estimates. The authors develop a robust framework combining a modified continued fraction, a topological model $A_\alpha$, and a marked renormalisation tower to explain degeneration phenomena of Siegel disk boundaries as $\alpha$ varies, with wide implications for rational maps and Julia sets. Overall, the paper provides a precise topological and dynamical description of the post-critical set for a broad class of maps, linking arithmetic properties to global topological outcomes and advancing renormalisation-based understanding of irrationally indifferent attractors.

Abstract

We study the post-critical set of a class of holomorphic systems with an irrationally indifferent fixed point. We prove a trichotomy for the topology of the post-critical set based on the arithmetic of the rotation number at the fixed point. The only options are Jordan curves, a one-sided hairy Jordan curves, and Cantor bouquet. This explains the degeneration of the closed invariant curves inside the Siegel disks, as one varies the rotation number.

Topology of irrationally indifferent attractors

TL;DR

This work classifications the local attractors of holomorphic maps with irrationally indifferent fixed points, proving a trichotomy for the post-critical set based on the arithmetic of the rotation number : a Jordan curve (Herman type), a one-sided hairy Jordan curve (Brjuno but not Herman), or a Cantor bouquet (non-Brjuno). Central to the analysis is a toy topological model that captures the essence of near-parabolic renormalisation and the role of arithmetic types, together with a thorough comparison between the toy model and the analytic renormalisation via explicit coordinate changes and contraction estimates. The authors develop a robust framework combining a modified continued fraction, a topological model , and a marked renormalisation tower to explain degeneration phenomena of Siegel disk boundaries as varies, with wide implications for rational maps and Julia sets. Overall, the paper provides a precise topological and dynamical description of the post-critical set for a broad class of maps, linking arithmetic properties to global topological outcomes and advancing renormalisation-based understanding of irrationally indifferent attractors.

Abstract

We study the post-critical set of a class of holomorphic systems with an irrationally indifferent fixed point. We prove a trichotomy for the topology of the post-critical set based on the arithmetic of the rotation number at the fixed point. The only options are Jordan curves, a one-sided hairy Jordan curves, and Cantor bouquet. This explains the degeneration of the closed invariant curves inside the Siegel disks, as one varies the rotation number.

Paper Structure

This paper contains 39 sections, 56 theorems, 234 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Irrationally indifferent attractors
  3. Statements of the results
  4. Methodology
  5. Arithmetic classes of Brjuno and Herman
  6. Modified continued fraction
  7. Brjuno numbers
  8. Herman numbers
  9. Topological model for the post-critical set
  10. Model for the changes of coordinates
  11. Successive changes of coordinates
  12. The straight topological model
  13. Hairy Cantor sets and Cantor bouquets
  14. Topology of the model
  15. Dynamics on the topological model
  16. ...and 24 more sections

Key Result

Theorem A

There is $N\geq 2$ such that for every $\alpha \in \mathrm{HT}_{N}$ and every $f(z)=e^{2\pi i \alpha} z+ O(z^2)$ in the Inou-Shishikura class ${\mathcal{F}}$, one of the following holds: The trichotomy also holds for the quadratic polynomials $P_\alpha$, when $\alpha \in \mathrm{HT}_{N}$.

Figures (11)

  • Figure 1: Left image: computer simulations of the orbit of $c_{P_\alpha}$ for rotation numbers $\alpha=[2,2,\overline{2}]$, $[2,2,10^2, \overline{2}]$, $[2,2,10^4, \overline{2}]$, and $[2,2,10^8, \overline{2}]$. Right image: computer simulations of the orbit of $c_{P_\alpha}$ for $\alpha=[2,2,\overline{2}]$, $[2,2,10^2, \overline{2}]$, and $[2,2,10^2, 10^8, \overline{2}]$.
  • Figure 2: Conjugacy of the non-autonomous dynamics of the changes of coordinates in two renormalisation schemes. The map $U_0$ projects to $U:\Lambda(c_f) \cup \Delta(f) \to \hat{A}_\alpha$ which conjugates $f$ on $\Lambda(c_f)$ to $T_\alpha$ on $\partial \hat{A}_\alpha$.
  • Figure 3: The black curves are the images of some horizontal lines by $Y_r$. The vertical lines in blue, from left to right, are the images of the vertical lines $\operatorname{Re} w=-1$, $\operatorname{Re} w=0$, and $\operatorname{Re} w=1/\alpha$, under $Y_r$. Here, $r=1/(10+1/(1+1/(1+\dots)))$.
  • Figure 4: In the left hand column $\varepsilon_n=-1$ and in the right hand column $\varepsilon_n=+1$. The sets $K_n^0$ and $J_n^0$ are on the lower row, and the set $M_{n-1}^1$ is on the upper row.
  • Figure 5: The domain ${\mathcal{P}}_h$ and the special points associated to some $h\in\mathcal{I}\space\mathcal{S}_\alpha$. The alternating coloured croissants are the pre-images of vertical strips of width one by $\Phi_h$.
  • ...and 6 more figures

Theorems & Definitions (109)

  • Theorem A: trichotomy of irrationally indifferent attractors
  • Theorem B: degeneration of closed invariant curves
  • Theorem C: invariant sets in irrationally indifferent attractors
  • Corollary D
  • Corollary E
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 3.1
  • Corollary 3.2
  • ...and 99 more