2 results
Equidistribution for matings of quadratic maps with the modular group
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 3 / March 2024
- Published online by Cambridge University Press:
- 12 May 2023, pp. 859-887
- Print publication:
- March 2024
-
- 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.
Dynamics of hyperbolic correspondences
- Part of
-
- Journal:
- Ergodic Theory and Dynamical Systems / Volume 42 / Issue 8 / August 2022
- Published online by Cambridge University Press:
- 04 May 2021, pp. 2661-2692
- Print publication:
- August 2022
-
- Article
- Export citation
-
This paper establishes the geometric rigidity of certain holomorphic correspondences in the family
$(w-c)^q=z^p$
, whose post-critical set is finite in any bounded domain of 
$\mathbb {C}$
. In spite of being rigid on the sphere, such correspondences are J-stable by means of holomorphic motions when viewed as maps of 
$\mathbb {C}^2$
. The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.
