Limits of Quadratic Rational Maps: The Cantor Locus

TL;DR

The paper develops a detailed framework linking the Cantor locus boundary in the moduli space ${\bf rat}_2$ to parabolic dynamics via dynamical markings rel $P$ and rel $P_\lambda$. It shows that a partial Parametrization of the parabolic boundary arises from markings carried along the Cantor locus, yielding stability results: for $\lambda_k\to\omega$ subhorocyclicly, markings with fixed $x$ converge to a unique boundary parameter in $\mathcal{R}^{\omega}\setminus\mathcal{D}^{\omega}$; if the marking lies in the $p/q$-star, the parameter tends to infinity and a rescaled limit $G_T$ appears on the boundary of $\mathcal{R}^1$. The work also builds a model space $\Delta^{\lambda *}$ recording markings, and develops a detailed star- and twig-based toolkit (wires, petals, moduli) to analyze which wires land where and how critical values sit within the attracting basin. The combination of normal-form analysis, modulus estimates, and Fatou/linearizing coordinates yields a precise description of the boundary dynamics and rescaling phenomena, with a canonical link between the parabolic model $P_{\omega}$ and the rescaled $P_1$-type dynamics. This advances understanding of the global geometry of ${\bf rat}_2$ near the Cantor locus boundary and clarifies how boundary maps arise as limits of Cantor-locus sequences.

Abstract

The \emph{Cantor locus} is the unique hyperbolic component, in the moduli space of quadratic rational maps ${\bf rat}_2$, consisting of maps with totally disconnected Julia sets. Whereas the geometry and dynamics of the Cantor locus is well understood, its boundary and the dynamics of the maps on the boundary are not. In this paper, we explore the dynamics near the parabolic parts of the boundary. We introduce the concept \emph{dynamical marking} of a map $g$, relative to the quadratic, parabolic polynomial $\mathrm{P}_{\opq}(z)={\opq} z+z^2$, with $\opq=e^{2πip/q}$. A dynamical marking $(x,ψ)$ of $g$ is a conjugacy $ψ$ between $\mathrm{P}_{\opq}$ (on its parabolic basin of 0) and $g$, which \emph{marks} the dynamical position of the critical values $v_1=ψ(-\frac{λ^2}{4})$, $v_2=ψ(x)$ of $g$. We construct a local parametrization of the Cantor locus, which parametrizes by dynamical marking, and use it to prove a form of \emph{stability} of dynamical marking. That is, for sequences in the Cantor locus, of fixed dynamical marking $x$ and such that the eigenvalue $λ_k$ of the attracting fixed point tends to $\opq$ \emph{subhorocyclicly}, either the sequence converges to the unique parabolic parameter in the boundary, which has a fixed point eigenvalue $\opq$ and which is marked by $x$ relative to $\mathrm{P}_{\opq}$. Or, the sequence tends to infinity in ${\bf rat}_2$, and certain representatives $G_{λ_k,a_k}$ have \emph{rescaled} limits in the boundary of the Cantor locus within ${\bf rat}_2$.

Figures (17)

• Figure 1: The left picture shows $A$, the attracting basin of 0, for $\mathrm{P}=\mathrm{P}_{1/3}$. The attracting flower $\mathcal{P}_{0}$ is shown in pink. The figure also illustrates components $B^j$ and sectors $S^j$. The right picture shows $\mathcal{A}_m^*$, with the removed sepals shown in dark grey, and the removed (extended) flower $\widehat{\mathcal{P}}_{1/q}$ shown in pink. Dynamical pictures are made by Arnaud Chéritat.
• Figure 2: Illustrates dynamical marking rel $\mathrm{P}$ for $p/q=1/3$. The left shows the dynamical plane of a $g\in \Gamma_{\omega_{1/3},\sigma}$ (on the Riemann sphere), and the filled-in Julia set of $\mathrm{P}$ is on the right. The figure shows the domain $\mathcal{U}(x)$ and range $V$ of $\psi$ in pink, the marking $x\in\mathcal{A}^*$ of $g$ and the two critical values $v_1$ and $v_2$ of $g$. Dynamical pictures are made by Arnaud Chéritat.
• Figure 3: The figure illustrates $\mathbb{D}(-r+2\pi ip/q,r)$, $D_M$ and $\rho_x$ for $p/q=1/3$ and $M=1/3$.
• Figure 4: The figure illustrates the map $h_1$ (for $p/q=1/3$), with the augmented basin $\mathcal{A}$ of $\mathrm{P}$ on the left and the augmented basin $\widetilde{A}_1$ of $\mathrm{P}_1$ on the right (to the level $n=2$ for $\widetilde{A}_1$). $h_1$ maps the sets $C(z_0)$ to $h_1(z_0)$. The figure also illustrates $h_1(\mathrm{P}^{q-1}(x))$, cf. Theorem \ref{['thm:convergenceunbound']}. Dynamical pictures by Arnaud Chéritat.
• Figure 5: The figure illustrates $r_\lambda$, $\log\lambda$, $L$ and $\theta$.

Theorems & Definitions (43)

• Definition 1: Dynamical marking relative to $\mathrm{P}$
• Proposition 1
• Definition 2: Definition of a partial parameter $\mathbf{\Phi}$ on $\mathcal{R}$
• Theorem 1
• Proposition 2
• Corollary 1
• Theorem 2
• Lemma 1: eps00
• Definition 3: The $(\log\lambda,p/q)$-star of $A_{\lambda,\sigma}$ for $g_{\lambda,\sigma}$.
• Lemma 2