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Equidistribution for matings of quadratic maps with the Modular group

Vanessa Matus de la Parra
摘要
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 betweenthe modular group $\operatorname{PSL}_2(\mathbb{Z})$ and a quadratic rationalmap. We show for every $a\in\mathcal{K}$, the iterated images and preimagesunder $\mathcal{F}_a$ of nonexceptional points equidistribute, in spite of thefact that $\mathcal{F}_a$ is weakly-modular in the sense of Dinh, Kaufmann andWu but it is not modular. Furthermore, we prove that periodic pointsequidistribute as well.
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