Equidistribution for matings of quadratic maps with the Modular group

TL;DR

This paper demonstrates that for a family of holomorphic correspondences related to quadratic maps and the modular group, both iterated images/preimages of nonexceptional points and periodic points become uniformly distributed over time.

Contribution

It establishes equidistribution results for these correspondences, even though they are only weakly modular and not fully modular, extending understanding of their dynamical behavior.

Findings

Iterated images and preimages of nonexceptional points equidistribute.

Periodic points also exhibit equidistribution.

Results hold despite the correspondences being only weakly modular.

Abstract

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.