Abstract
We develop a 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) ([2]). 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.
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