Oral Exam

Equidistribution for matings of rational maps and the modular group

Vanessa Matus de la Parra

Thursday, June 8th, 2023
9:00 AM - 10:00 AM
Hylan 1106A & Zoom ID 918 227 3330

Equidistribution has been studied in several different settings in dynamical systems since Brolin’s paper in the 60’s. The most popular question is about the existence of a Borel probability measure \(\mu\) for our dynamical system \(F\) satisfying that for all but finitely many points \(z_0\), we have

\(\frac{1}{d(F)^n}(F^n)_*\delta_{z_0}\rightarrow\mu\), as \(n\to\infty\).

Here \((F^n)_*\) denotes the push-forward operator associated with \(F^n\), \(d(F)\) denotes the topological degree of \(F\), and the convergence is in the weak topology. Perhaps the most known of these settings is the case of rational maps on the Riemann sphere, which was proven simultaneously by Freire-Lopes-Mañé and Lyubich in the 80’s. There are plenty generalizations, for example for higher dimension. A different path to take is to consider multivalued maps. This is the case of holomorphic correspondences on \(\widehat{\mathbb{C}}\). These multivalued maps come from effective analytic cycles of pure dimension 1 having no fiber of the canonical projections. Bullett-Lomonaco proved that a given 1-parameter family of holomorphic correspondences \(\{ \mathcal{F}_a\}_{a\in\mathcal{K}}\) consists of “matings” between quadratic parabolic rational maps and the modular group . This mating structure helps us bring the result of Freire-Lopes-Mañé and Lyubich to this setting. In this talk, we will define and give an overview of this family, and then we will talk about its equidistribution of images, pre-images, and periodic points.

Event contact: vmatusde at ur dot rochester dot edu