Equidistribution for matings of quadratic maps with the modular group

Abstract

We study the asymptotic behavior of the family of holomorphic correspondences $\lbrace \mathcal {F}_a\rbrace _{a\in \mathcal {K}}$ , given by $$ \begin{align*}\bigg(\frac{az+1}{z+1}\bigg)^2+\bigg(\frac{az+1}{z+1}\bigg)\bigg(\frac{aw-1}{w-1}\bigg)+\bigg(\frac{aw-1}{w-1}\bigg)^2=3.\end{align*} $$ It was proven by Bullet and Lomonaco [Mating quadratic maps with the modular group II. Invent. Math.220(1) (2020), 185–210] that $\mathcal {F}_a$ is a mating between the modular group $\operatorname {PSL}_2(\mathbb {Z})$ and a quadratic rational map. We show for every $a\in \mathcal {K}$ , the iterated images and preimages under $\mathcal {F}_a$ of non-exceptional points equidistribute, in spite of the fact that $\mathcal {F}_a$ is weakly modular in the sense of Dinh, Kaufmann, and Wu [Dynamics of holomorphic correspondences on Riemann surfaces. Int. J. Math.31(05) (2020), 2050036], but it is not modular. Furthermore, we prove that periodic points equidistribute as well.