Related papers: Matings with laminations

We introduce the notion of n-mating in this work, which includes the classical mating of polynomials as a special case. The new notion brings further links between the polynomial world and the rational world than the classical one, as well…

In this Note, we present recent developments in the Renormalization Theory of quadratic polynomials and discuss their applications, with an emphasis on the MLC conjecture, the problem of local connectivity of the Mandelbrot set, and on its…

The aim of this work is to describe the equivalence relations in $\Q/\Z$ that arise as the rational lamination of polynomials with all cycles repelling. We also describe where in parameter space one can find a polynomial with all cycles…

Suppose that a $X$ is an \emph{unshielded} plane continuum (i.e., $X$ coincides with the boundary of the unbounded complementary component of $X$). Then there exists a \emph{finest monotone} map $m:X\to L$, where $L$ is a locally connected…

We will show the Mandelbrot set $M$ is locally conformally inhomogeneous: the only conformal map $f$ defined in an open set $U$ intersecting $\partial M$ and satisfying $f(U\cap\partial M)\subset \partial M$ is the identity map. The proof…

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…

We characterize 2-dimensional complexes associated canonically with basis graphs of matroids as simply connected triangle-square complexes satisfying some local conditions. This proves a version of a (disproved) conjecture by Stephen Maurer…

Little is known about the global structure of the basins of attraction of Newton's method in two or more complex variables. We make the first steps by focusing on the specific Newton mapping to solve for the common roots of $P(x,y) =…

This paper focuses on local curvature invariants associated with bi-Lagrangian structures. We establish several geometric conditions that determine when the canonical connection is flat, building on our previous findings regarding…

Consider a quadratic rational self-map of the Riemann sphere such that one critical point is periodic of period 2, and the other critical point lies on the boundary of its immediate basin of attraction. We will give explicit topological…

We prove that topologically conjugate non-renormalizable polynomials are quasi-conformally conjugate. From this we derive that each such polynomial can be approximated by a hyperbolic polynomial. As a by product we prove that the Julia set…

In the field of the Jacobian conjecture it is well-known after Druzkowski that from a polynomial "cubic-homogeneous" mapping we can build a higher-dimensional "cubic-linear" mapping and the other way round, so that one of them is invertible…

To investigate the degree $d$ connectedness locus, Thur\-ston studied \emph{$\sigma_d$-invariant laminations}, where $\sigma_d$ is the $d$-tupling map on the unit circle, and built a topological model for the space of quadratic polynomials…

In this paper, we give a family of rational maps whose Julia sets are Cantor circles and show that every rational map whose Julia set is a Cantor set of circles must be topologically conjugate to one map in this family on their…

We study moduli spaces $\mathcal{M}$ of holomorphic maps $U$ from Riemann surfaces to $\mathbb{R}^{4}$ with boundaries on the Lagrangian cylinder over a Legendrian link $\Lambda \subset (\mathbb{R}^{3}, \xi_{std})$. We allow our domains,…

We give an alternate proof to the following generalization of the uniformization theorem by Bonk and Kleiner. Any linearly locally connected and Ahlfors 2-regular closed metric surface is quasisymmetrically equivalent to a model surface of…

The moduli space ${\rm M}_{d}$, of complex rational maps of degree $d \geq 2$, is a connected complex orbifold which carries a natural real structure, coming from usual complex conjugation. Its real points are the classes of rational maps…

In this paper we prove the following: Take any "small Mandelbrot set" and zoom in a neighborhood of a parabolic or Misiurewicz parameter in it, then we can see a quasiconformal image of a Cantor Julia set which is a perturbation of a…

This article discusses some topological properties of the dynamical plane ($z$-plane) of the holomorphic family of meromorphic maps $\lambda + \tan z^2$ for $ \lambda \in \mathbb C$. In the dynamical plane, we prove that there is no Herman…

We investigate the discontinuity of codings for the Julia set of a quadratic map. To each parameter ray, we associate a natural coding for Julia sets on the ray. Given a hyperbolic component $H$ of the Mandelbrot set, we consider the…