Dynamics of Modular Matings

TL;DR

This paper develops a dynamical theory for holomorphic correspondences that are matings between the modular group and parabolic rational maps, analyzing their geometric and combinatorial properties.

Contribution

It introduces bi-infinite coding sequences for geodesics, proves landing theorems, and establishes a stronger Yoccoz inequality for these correspondences.

Findings

Connectedness locus is contained in a specific lune in parameter space.

Landing theorems for periodic and preperiodic geodesics are proved.

A stronger Yoccoz inequality for fixed points is established.

Abstract

We develop dynamical theory for the family of holomorphic correspondences $\mathcal{F}_a$ proved by the current authors to be matings between the modular group and parabolic rational maps in the Milnor slice $Per_1(1)$ (in 'Mating quadratic maps with the modular group II'). Such a mating endows the complement of the limit set of $\mathcal{F}_a$ with the geometry of the hyperbolic plane, equipped with the action of the modular group. We introduce bi-infinite coding sequences for geodesics in this complement, utilising continued fraction expressions of end points; we prove landing theorems for periodic and preperiodic geodesics, and we establish a stronger Yoccoz inequality for repelling fixed points of these correspondences than Yoccoz's classical inequality for quadratic polynomials. We deduce that the connectedness locus of the family $\mathcal{F}_a$ is contained in a particular lune in parameter space.