  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.
