Dynamics of Modular Matings
Abstract
We develop dynamical theory for the family of holomorphic correspondences Fa proved by the current authors to be matings between the modular group and parabolic rational maps in the Milnor slice Per1(1) (in 'Mating quadratic maps with the modular group II'). Such a mating endows the complement of the limit set of Fa 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 Fa is contained in a particular lune in parameter space.