Dynamics of Modular Matings

Abstract

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 $(\log q)/q^2$