Equidistribution for matings of quadratic maps with the Modular group

Abstract

We study the asymptotic behavior of the family of holomorphic correspondences $\{{\mathcal{F}}_a{\}}_{a\in \mathcal{K}}$, given by $${\left(\frac{az+1}{z+1}\right)}^2+\left(\frac{az+1}{z+1}\right)\left(\frac{aw-1}{w-1}\right)+{\left(\frac{aw-1}{w-1}\right)}^2=3.$$ It was proven by Bullet and Lomonaco that ${\mathcal{F}}_a$ is a mating between the modular group ${\mathrm{PSL}}_2(\mathbb{Z})$ and a quadratic rational map. We show for every $a\in \mathcal{K}$, the iterated images and preimages under ${\mathcal{F}}_a$ of nonexceptional points equidistribute, in spite of the fact that ${\mathcal{F}}_a$ is weakly-modular in the sense of Dinh, Kaufmann and Wu but it is not modular. Furthermore, we prove that periodic points equidistribute as well.

Cited by in corpus (1)

• Equidistribution of graphs of holomorphic correspondences