arXiv subjects

Miguel Ratis Laude

Publications and source records attributed to Miguel Ratis Laude.

4 recordsLinked to original sources

arXiv2026

Tessellating the discreteness locus for the modular mating family of correspondences

The modular Mandelbrot set $M_Γ$, the connectedness locus of the modular mating family of 2 : 2 holomorphic correspondences $\mathcal{F}_a$ on the Riemann sphere, is homeomorphic to the classical Mandelbrot set $M$. The Klein combination locus $\mathcal{K}$ (the "discreteness locus" of the family $\mathcal{F}_a$) is a pinched neighborhood of $M_Γ$ in the $a$-plane, pinched at the root point. We construct a canonical map $Ψ$ from $\mathcal{K}\setminus M_Γ$ into the hyperbolic plane $\mathbb{H}$, inspired by the construction of Douady and Hubbard for their celebrated conformal bijection $Φ: \mathbb{C}\setminus M \to \mathbb{C}\setminus \overline{\mathbb{D}}$, and we prove that $Ψ$ is analytic. This map $Ψ$ induces a tessellation of $\mathcal{K}\setminus M_Γ$ by pulling back a tessellation of $\mathbb{H}$ invariant under the modular group. We develop a series of conjectures concerning the structure of $\mathcal{K}$, its boundary, and $Ψ(\mathcal{K}) \subset \mathbb{H}$.

math.DS↗

arXiv2026

Matings between compositions of rational maps and free products of finite cyclic groups

Given a pair of rational maps $(f, g)$, of degrees $p$ and $q$, each with a parabolic fixed point having a fully invariant simply-connected basin of attraction, we construct an algebraic correspondence $F$ on the Riemann sphere, of bidegree $(pq, pq)$, realizing a mating between the two compositions $g\circ f$ and $f\circ g$ of the maps, and the parabolic faithful discrete representation of the free product of cyclic groups of orders $p + 1$ and $q + 1$. We also show that $F$ is the composition of a pair of deleted covering correspondences of rational maps which are conjugated to polynomials of degrees $p + 1$ and $q + 1$. We generalize our method to construct matings between compositions of pairs of polynomials and (non-parabolic) faithful Kleinian representations of the same group, now with connected regular set. As far as we are aware, these matings between pairs of maps and groups are the first examples that are not time-reversible (that is, they are not conjugate to their own inverses).

math.DS↗

arXiv2025

David regularity of the Yoccoz extension

A central problem in the study of critical circle dynamics is understanding the regularity of Yoccoz conjugators - circle homeomorphisms that conjugate critical circle maps with irrational rotation numbers to their corresponding rigid rotations. One can approach this problem from a different angle by studying the regularity of extensions of these maps to the unit disk. Of particular interest is the question of when such a conjugator admits a David extension. Building on the work of Petersen and Zakeri, we classify the David regularity of a specific extension process known as the Yoccoz extension.

math.DS↗

arXiv2025

Continuity of matings of Kleinian groups and polynomials

In recent years, the study of holomorphic correspondences as dynamical systems that can display behaviors of both rational maps and Kleinian groups has gained a good amount of attention. This phenomenon is related to the Sullivan dictionary, a list of parallels between the theories of these two systems. We build upon a surgical construction of such matings, due to Bullett and Harvey, increasing the degree of maps we consider, and proving regularity properties of the mating map on parameter spaces: namely, analyticity on the interior of its domain of definition, and continuity under quasiconformal rigidity on the boundary.

math.DS↗