1 results
Equidistribution for matings of quadratic maps with the modular group
- Part of
- Journal: Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press: 12 May 2023, pp. 1-29
-
- Article
-
- You have access
- Open access
- HTML
- Export citation
-
We study the asymptotic behavior of the family of holomorphic correspondences
$\lbrace \mathcal {F}_a\rbrace _{a\in \mathcal {K}}$
, 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
$$ \begin{align*}\bigg(\frac{az+1}{z+1}\bigg)^2+\bigg(\frac{az+1}{z+1}\bigg)\bigg(\frac{aw-1}{w-1}\bigg)+\bigg(\frac{aw-1}{w-1}\bigg)^2=3.\end{align*} $$
$\mathcal {F}_a$
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.
