Equidistribution for matings of quadratic maps with the modular group

We study the asymptotic behavior of the family of holomorphic correspondences, given by It was proven by Bullet and Lomonaco [Mating quadratic maps with the modular group II. Invent. Math. 220(1) (2020), 185-210] that is a mating between the modular group and a quadratic rational map. We show for every, the iterated images and preimages under of non-exceptional points equidistribute, in spite of the fact that 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.