We study the asymptotic behavior of the family of holomorphic correspondences $\lbrace\mathcal{ F}_a\rbrace_{ a\in\mathcal{ K}}$, 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 $\operatorname{ PSL}_2(\mathbb{ Z})$ and a quadratic rational map. We show for every $a\in\mathcal{ K}$, the iterated images and preimages under $\mathcal{ F}_a$ of 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.
Categories: math.DS