Limits of Quadratic Rational Maps: The Cantor Locus

TL;DR

This paper investigates the boundary dynamics of the Cantor locus in quadratic rational maps, introducing dynamical marking to understand stability and limits near parabolic boundary points.

Contribution

It introduces the concept of dynamical marking for maps near parabolic boundaries and constructs a local parametrization to analyze stability and limits.

Findings

Sequences with fixed dynamical marking converge to boundary parabolic maps.

Sequences with eigenvalues tending to a root of unity tend to infinity or converge to boundary maps.

The paper establishes stability results for dynamical markings near the boundary.

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 $\opq=e^{2\pi ip/q}$. A dynamical marking $(x,\psi)$ of $g$ is a conjugacy $\psi$ between $\mathrm{P}_{\opq}$ (on its parabolic basin of 0) and $g$, which \emph{marks} the dynamical position of the critical values $v_1=\psi(-\frac{\lambda^2}{4})$, $v_2=\psi(x)$ 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 $\lambda_k$ 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 $G_{\lambda_k,a_k}$ have \emph{rescaled} limits in the boundary of the Cantor locus within ${\bf rat}_2$.