arXiv subjects

Shaun Bullett

Publications and source records attributed to Shaun Bullett.

8 recordsLinked to original sources

arXiv2026

Tessellating the discreteness locus for the modular mating family of correspondences

The modular Mandelbrot set $M_\Gamma$, 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 $M_\Gamma$ in the $a$-plane, pinched at the root point. We construct a canonical map $\Psi$ from $\mathcal{K}\setminus M_\Gamma$ into the hyperbolic plane $\mathbb{H}$, inspired by the construction of Douady and Hubbard for their celebrated conformal bijection $\Phi: \mathbb{C}\setminus M \to \mathbb{C}\setminus \overline{\mathbb{D}}$, and we prove that $\Psi$ is analytic. This map $\Psi$ induces a tessellation of $\mathcal{K}\setminus M_\Gamma$ 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 $\Psi(\mathcal{K}) \subset \mathbb{H}$.

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↗

arXiv2024

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.

math.DS↗

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 $\mathcal{M}$.

math.DS↗

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.

math.DS↗

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 ${\bf f}_c$.

math.DS↗

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 $Per_1(1)$.

math.DS↗

arXiv2005

Pinching Holomorphic Correspondences

For certain classes of holomorphic correspondences which are matings between Kleinian groups and polynomials, we prove the existence of pinching deformations, analogous to Maskit's deformations of Kleinian groups which pinch loxodromic elements to parabolic elements. We apply our results to establish the existence of matings between quadratic maps and the modular group, for a large class of quadratic maps, and of matings between the quadratic map $z\to z^2$ and circle-packing representations of the free product $C\_2*C\_3$ of cyclic groups of order 2 and 3.

math.DS↗