Dynamics of modular matings

We develop a dynamical theory for the family of holomorphic correspondences Fa proved by current authors to be matings between modular group and parabolic rational maps in Milnor slice Per1(1) ([2]). Such mating endows complement limit set with geometry hyperbolic plane, equipped action group. introduce bi-infinite coding sequences geodesics this complement, utilising continued fraction expressions end points; we prove landing theorems periodic preperiodic geodesics, establish stronger Yoccoz inequality repelling fixed points these than Yoccoz's classical quadratic polynomials. deduce that connectedness locus is contained particular lune parameter space.