Dynamics of Modular Matings

In the paper 'Mating quadratic maps with the modular group II' the current authors proved that each member of the family of holomorphic $(2:2)$ correspondences $\mathcal{F}_a$: $$\left(\frac{az+1}{z+1}\right)^2+\left(\frac{az+1}{z+1}\right)\left(\frac{aw-1}{w-1}\right) +\left(\frac{aw-1}{w-1}\right)^2=3,$$ introduced by the first author and C. Penrose in 'Mating quadratic maps with the modular group', is a mating between the modular group and a member of the parabolic family of quadratic rational maps $P_A:z\to z+1/z+A$ whenever the limit set of $\mathcal{F}_a$ is connected. Here we provide a dynamical description for the correspondences $\mathcal{F}_a$ which parallels the Douady and Hubbard description for quadratic polynomials. We define a B\"ottcher map and a Green's function for $\mathcal{F}_a$, and we show how in this setting periodic geodesics play the role played by external rays for quadratic polynomials. Finally, we prove a Yoccoz inequality which implies that for the parameter $a$ to be in the connectedness locus $M_{\Gamma}$ of the family $\mathcal{F}_a$, the value of the log-multiplier of an alpha fixed point which has combinatorial rotation number $1/q$ lies in a strip whose width goes to zero at rate proportional to $(\log q)/q^2$

Pinching Holomorphic Correspondences

International audienceFor certain classes of holomorphic correspondences which are matings between Kleinian groups and polynomials, we prove the existence of pinching deformations, analogous to Maskit's deformations of Kleinian groups which pinch loxodromic elements to parabolic elements. We apply our results to establish the existence of matings between quadratic maps and the modular group, for a large class of quadratic maps, and of matings between the quadratic map $z\to z^2$ and circle-packing representations of the free product $C_2*C_3$ of cyclic groups of order $2$ and $3$

A Gallery of Iterated Correspondences

this paper we summarise our general results concerning iterated (2; 2) correspondences on the Riemann sphere, survey the examples we have constructed so far---in particular, "matings" between Kleinian group and rational map actions---and indicate some directions for further study. Our motivation for undertaking this investigation was the striking series of results of Sullivan obtained by applying quasiconformal deformation theory to both rational maps and Kleinian groups [Sullivan 1984; 1985a; 1985b]. Our hope was that by studying iterated correspondences we could obtain further insight into how these classes of dynamical systems are related. The results outlined in this paper are a step in that direction: we believe the examples also have considerable interest in their own right. 2. QUADRATIC CORRESPONDENCES AND THEIR GRAPH