Bers Slices in Families of Univalent Maps

Advances in Mathematics, 2021

In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in . Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns these reflections to parabolic rational maps (which is the crucial technical ingredient of our theory). It induces a straightening map between the parameter plane of Schwarz reflections and the parabolic Tricorn. We describe various properties of this straightening highlighting the issues related to its anti-holomorphic nature. We complete the discussion by comparing our family with the classical Bullett-Penrose family of matings between groups and rational maps induced by holomorphic correspondences. More precisely, we show that the Schwarz reflections give rise to anti-holomorphic correspondences that are matings of parabolic rational maps with the modular group.

Conformal Geometry and Dynamics of the American Mathematical Society

According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group H H whose limit set is a generalized Apollonian gasket Λ H \Lambda _H . We design a surgery that relates H H to a rational map g g whose Julia set J g \mathcal {J}_g is (non-quasiconformally) homeomorphic to Λ H \Lambda _H . We show for a large class of triangulations, however, the groups of quasisymmetries of Λ H \Lambda _H and J g \mathcal {J}_g are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of H H , this group is equal to the group of Möbius symmetries of Λ H \Lambda _H , which is the semi-direct product of H H itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when Λ H \Lambda _H is the classical Apollonian gasket), we give a quasiregular model for the above actions whi...

Journal of Difference Equations and Applications, 2016

We consider the family of rational maps given by F λ (z) = z n +λ/z d where n, d ∈ N with 1/n + 1/d < 1, the variable z ∈ C and the parameter λ ∈ C. It is known [1] that when n = d ≥ 3 there are n − 1 small copies of the Mandelbrot set symmetrically located around the origin in the parameter λ−plane. These baby Mandelbrot sets have "antennas" attached to the boundaries of Sierpiński holes. Sierpiński holes are open simply connected subsets of the parameter space for which the Julia sets of F λ are Sierpiński curves. In this paper we generalize the symmetry properties of F λ and the existence of the n − 1 baby Mandelbrot sets to the case when 1/n + 1/d < 1 where n is not necessarily equal to d.

Geometric and Functional Analysis

In a previous paper [LLM20], we constructed an explicit dynamical correspondence between certain Kleinian reflection groups and certain anti-holomorphic rational maps on the Riemann sphere. In this paper, we show that their deformation spaces share many striking similarities. We establish an analogue of Thurston's compactness theorem for critically fixed anti-rational maps. We also characterize how deformation spaces interact with each other and study the monodromy representations of the union of all deformation spaces. Contents 1. Introduction 1 2. Degeneration of anti-Blaschke products 8 3. Realization of (d + 1)-ended ribbon trees 19 4. Boundedness and mutual interaction of deformation spaces 29 5. Markov partitions and monodromy representations 39 Appendix A. Laminations, automorphisms and accesses 42 Appendix B. Shared matings, self-bumps and disconnected roots 47 References 52

International Journal of Mathematics and Mathematical Sciences, 2008

The classical Schwarz-Christoffel formula gives conformal mappings of the upper half-plane onto domains whose boundaries consist of a finite number of line segments. In this paper, we explore extensions to boundary curves which in one sense or another are made up of infinitely many line segments, with specific attention to the "infinite staircase" and to the Koch snowflake, for both of which we develop explicit formulas for the mapping function and explain how one can use standard mathematical software to generate corresponding graphics. We also discuss a number of open questions suggested by these considerations, some of which are related to differentials on hyperelliptic surfaces of infinite genus.

Forum of Mathematics, Sigma

In this article, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs, has several dynamical consequences. As an application of this correspondence, we give complete answers to geometric mating problems for critically fixed anti-rational maps.

2011

In this paper, we consider the family of rational maps

2016

Cambridge University Press eBooks, 2010

The residual Julia set, denoted by J r (f), is defined to be the subset of those points of the Julia set which do not belong to the boundary of any component of the Fatou set. The points of J r (f) are called buried points of J(f) and a component of J(f) which is contained in J r (f) is called a buried component. In this paper we survey the most important results related with the residual Julia set for several classes of functions. We also give a new criterium to deduce the existence of buried points and, in some cases, of unbounded curves in the residual Julia set (the so called Devaney hairs). Some examples are the sine family, certain meromorphic maps constructed by surgery and the exponential family.

Constructive Approximation, 1994

ABSTRACT Let C be a simply connected domain, 0, and let n,nN, be the set of all polynomials of degree at mostn. By n() we denote the subset of polynomials p n withp(0)=0 andp(D), whereD stands for the unit disk {z: |z|we denote the maximal range of these polynomials. Letf be a conformal mapping fromD onto ,f(0)=0. The main theme of this note is to relate n (or some important aspects of it) to the imagesf s (D), wheref s (z):=f[(1–s)z], 0sc 0 such that, forn2c 0,