arXiv subjects

Luna Lomonaco

Publications and source records attributed to Luna Lomonaco.

14 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 $Ψ(\mathcal{K}) \subset \mathbb{H}$.

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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).

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arXiv2026

Mating parabolic rational maps with Hecke groups

We prove that any degree $d$ rational map having a parabolic fixed point of multiplier $1$ with a fully invariant and simply connected immediate basin of attraction is mateable with the Hecke group $H_{d+1}$, with the mating realized by an algebraic correspondence. This confirms the parabolic version of a conjecture on mateability between rational maps and Hecke groups made in \cite{BF1}. The proof is in two steps. The first is the construction of a pinched polynomial-like map which is a mating between a parabolic rational map and a parabolic circle map associated to the Hecke group. The second is lifting this pinched polynomial-like map to an algebraic correspondence via a suitable branched covering.

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arXiv2025

Algebraic correspondences and Schwarz reflections: Where rational dynamics meets Kleinian groups

We present an overview of the rapidly evolving field of dynamics of algebraic correspondences, with a focus on matings between rational maps and Kleinian groups. These correspondences exhibit rich dynamics, both within the Sullivan dictionary and beyond. We highlight unifying structures in their parameter spaces, showing how moduli spaces of rational maps and Kleinian groups naturally connect. We also outline a range of applications of the techniques developed in this framework and conclude with several promising directions.

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arXiv2025

On qc compatibility of satellite copies of the Mandelbrot set: II

The Mandelbrot set is a fractal which classifies the behaviour of complex quadratic polynomials. Although its remarkably simple definition: $\mathcal{M}:=\{c \in \mathbb{C}\,|\,Q_c(0)^n \nrightarrow \infty \mbox{ as } n\rightarrow \infty, \mbox{ where } Q_c(z)=z^2+c\}$, it is a central object in complex dynamics, and it has been charming and intriguing since it has first been defined and drawn. A fascinating fact is the presence of little copies of the Mandelbrot set in the Mandelbrot set itself (and in many other parameter planes). There exist two different kinds of little copies of the Mandelbrot set within the Mandelbrot set: the primitive copies, visually similar to the Mandelbrot set, with a cusp at the root of the principal hyperbolic component, and the satellite copies, whose principal hyperbolic component has no cusp, i.e. has smooth boundary across the root point. Lyubich proved that the primitive copies of $\mathcal{M}$ satisfy a stronger regularity condition: they are quasiconformally homeomorphic to $\mathcal{M}$. The satellite copies are homeomorphic to $\mathcal{M}$, but the homeomorphism is only quasiconformal outside neighbourhoods of the root. The question that remained open was: are the satellite copies mutually quasiconformally homeomorphic? In a previous work, we showed that the satellite copies with rotation numbers with different denominators are not quasiconformally homeomorphic. Here we complete the picture, by showing that, for any $q$, the satellite copies $\mathcal{M}_{p/q}$ and $\mathcal{M}_{p'/q}$ of the Mandelbrot set $\mathcal{M}$ with rotation numbers with the same denominator $q$ are quasiconformal homeomorphic.

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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.

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arXiv2023

Mating quadratic maps with the modular group III: The modular Mandelbrot set

We prove that there exists a homeomorphism $χ$ between the connectedness locus $\mathcal{M}_Γ$ for the family $\mathcal{F}_a$ of $(2:2)$ holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set $\mathcal{M}_1$. The homeomorphism $χ$ is dynamical ($\mathcal{F}_a$ is a mating between $PSL(2,\mathbb{Z})$ and $P_{χ(a)}$), it is conformal on the interior of $\mathcal{M}_Γ$, and it extends to a homeomorphism between suitably defined neighbourhoods in the respective one parameter moduli spaces. Following the recent proof by Petersen and Roesch that $\mathcal{M}_1$ is homeomorphic to the classical Mandelbrot set $\mathcal{M}$, we deduce that $\mathcal{M}_Γ$ is homeomorphic to $\mathcal{M}$.

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arXiv2022

Dynamics of Modular Matings

We develop dynamical theory for the family of holomorphic correspondences $\mathcal{F}_a$ proved by the current authors to be matings between the modular group and parabolic rational maps in the Milnor slice $Per_1(1)$ (in 'Mating quadratic maps with the modular group II'). Such a mating endows the complement of the limit set of $\mathcal{F}_a$ with the geometry of the hyperbolic plane, equipped with the action of the modular group. We introduce bi-infinite coding sequences for geodesics in this complement, utilising continued fraction expressions of end points; we prove landing theorems for periodic and preperiodic geodesics, and we establish a stronger Yoccoz inequality for repelling fixed points of these correspondences than Yoccoz's classical inequality for quadratic polynomials. We deduce that the connectedness locus of the family $\mathcal{F}_a$ is contained in a particular lune in parameter space.

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arXiv2020

A note on parabolic-like maps

We show that the definition of parabolic-like map can be slightly modified, by asking $\partial Δ$ to be a quasiarc out of the parabolic fixed point, instead of the dividing arcs to be $C^1$ on $[-1,0]$ and $[0,1]$.

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arXiv2018

A rigidity result for some parabolic germs

The goal of this article is to prove a rigidity result for unicritical polynomials with parabolic cycles. More precisely, we show that if two unicritical polynomials have conformally conjugate parabolic germs, then the polynomials are affinely conjugate.

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arXiv2017

Correspondences in complex dynamics

This paper surveys some recent results concerning the dynamics of two families of holomorphic correspondences, namely ${\mathcal F}_a:z \to w$ defined by the relation $$\left( \frac{aw-1}{w-1} \right)^2 + \left( \frac{aw-1}{w-1} \right) \left( \frac{az +1}{z+1} \right) + \left( \frac{az+1}{z+1} \right)^2 =3,$$ and $$\mathbf{f}_c(z)=z^β +c, \mbox{ where } 1<β=p/q \in \mathbb{Q},$$ which is the correspondence $\mathbf{f}_c:z \to w$ defined by the relation $$(w-c)^q=z^p.$$ Both can be regarded as generalizations of the family of quadratic maps $f_c(z)=z^2+c$. We describe dynamical properties for the family $\mathcal{F}_a$ which parallel properties enjoyed by quadratic polynomials, in particular a Böttcher map, periodic geodesics and Yoccoz inequality, and we give a detailed account of the very recent theory of holomorphic motions for hyperbolic multifunctions in the family ${\bf f}_c$.

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arXiv2017

Mating quadratic maps with the modular group II

In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of $(2:2)$ holomorphic correspondences $\mathcal{F}_a$: $$\left(\frac{aw-1}{w-1}\right)^2+\left(\frac{aw-1}{w-1}\right)\left(\frac{az+1}{z+1}\right) +\left(\frac{az+1}{z+1}\right)^2=3$$ and proved that for every value of $a \in [4,7] \subset \mathbb{R}$ the correspondence $\mathcal{F}_a$ is a mating between a quadratic polynomial $Q_c(z)=z^2+c,\,\,c \in \mathbb{R}$ and the modular group $Γ=PSL(2,\mathbb{Z})$. They conjectured that this is the case for every member of the family $\mathcal{F}_a$ which has $a$ in the connectedness locus. We prove here that every member of the family $\mathcal{F}_a$ which has $a$ in the connectedness locus is a mating between the modular group and an element of the parabolic quadratic family $Per_1(1)$.

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arXiv2016

On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I

In the paper 'On the dynamics of polynomial-like mappings' Douady and Hubbard introduced the notion of polynomial-like maps. They used it to identify homeomophic copies of the Mandelbrot set inside the Mandelbrot set. They conjectured that in case of primitive copies the homeomorphism between the homeomorphic copy of the Mandelbrot set and the Mandelbrot set is q.-c., and similarly in the satellite case, it is q.-c. off any small neighborhood of the root. These conjectures are now Theorems due to Lyubich. The satellite copies of the Mandelbrot set are clearly not q-c homeomorphic to the Mandelbrot set. But are they mutually q-c homeomorphic? Or even q-c homeomorphic to half of the logistic Mandelbrot set? In this paper we prove that, in general, the induced Douady-Hubbard homeomorphism is not the restriction of a q-c homeomorphism: For any two satellite copies of the Mandelbrot set, the induced Douady-Hubbard homeomorphism is not q-c, if the root multipliers, which are primitive q and q' roots of unity, have q different from q'.

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arXiv2016

On parabolic external maps

We prove that any $C^{1+BV}$ degree $d \geq 2$ circle covering $h$ having all periodic orbits weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. We use this to connect the space of parabolic external maps (coming from the theory of parabolic-like maps) to metrically expanding circle coverings.

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