Arithmetic Geometric Model for the Renormalisation of Bi-critical Irrationally Indifferent Attractors

TL;DR

This work introduces a two-parameter bi-critical renormalisation toy model for irrationally indifferent fixed points with two recurrent critical points. It builds a compact, star-like phase space $\mathds{M}_{[\alpha],[\beta]}$ and a two-parameter map $\mathds{T}_{[\alpha],[\beta]}$, together with a renormalisation operator $\mathcal{R}$ that acts as $([\alpha],[\beta])\mapsto([-1/\alpha],[-\beta/\alpha])$, capturing both arithmetical and geometric aspects of the dynamics. A central result is a topological trichotomy: the boundary $\mathds{A}_{[\alpha],[\beta]}$ is a Jordan curve if $\alpha$ is Herman (Brjuno-type), a one-sided hairy Jordan curve if $\alpha$ is Brjuno but not Herman, and a Cantor bouquet if $\alpha$ is not Brjuno, with the boundary type determined by the modified Brjuno sum $\mathcal{B}(\alpha,\beta)$. The paper also provides quantitative Siegel-disk size estimates and a rigorous justification of the model via Fatou coordinates, Ostrowski-type expansions, and connections to the uni-critical case, offering a robust framework for understanding bi-critical irrationals-indifferent attractors in holomorphic dynamics.

Abstract

In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points of holomorphic maps with two critical points. The model incorporates arithmetic properties of the rotation number at the fixed point, as well as the "angle" between the two critical points. Using this model for the renormalisation, we build a topological model for the local dynamics of such maps. We also explain the topology of the maximal invariant set for the model, and the dynamics of the map on the maximal invariant set.

Figures (10)

• Figure 1: Computations describing fatou coordinates for bi-critical cubics with a fixed $\alpha$ and $\beta=1/2$ on the left, and $\beta=0$ with the same $\alpha$ on the right. The white points are the critical points, the yellow repelling fixed points, red the fixed point at $0$, and orange represents the forward $\lfloor1/|\alpha|\rfloor$ images of a ray coming out of $0$ to one of the nearby repelling fixed points.
• Figure 2: The image of the same horizontal line for $G_{r,0}$ and $G_{r,1/2}$
• Figure 3: The $g_r(x+iy)$ functions are generated by the distances from the point $e^{-3\pi r}$ to a family of circles
• Figure 4: The $G_{r,s}$ function (represented in green), always lies between the space drawn by the two $g_r$ functions that generate the $a_{r,s}$ and $b_{r,s}$
• Figure 5: An image of the construction for $\epsilon_n=-1$. The set $I^1_{n-1}$ is pictured in the upper row.

Theorems & Definitions (118)

• Definition 1.1
• Definition 1.2
• Definition 1.3: Arithmetic of Orbits
• Theorem 1.4
• Theorem 1.5: Trichotomy
• Theorem 1.6: Conjugacy Classes
• Conjecture 1.7
• Remark 2.1
• Remark 2.2
• proof