# Shapes of polynomial Julia sets

@article{Lindsey2014ShapesOP, title={Shapes of polynomial Julia sets}, author={Kathryn A. Lindsey}, journal={Ergodic Theory and Dynamical Systems}, year={2014}, volume={35}, pages={1913 - 1924} }

Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by Julia sets of polynomials. Finite collections of disjoint Jordan domains can be approximated by the basins of attraction of rational maps.

#### 10 Citations

Shapes of polynomial Julia sets, revisited

- Mathematics
- 2016

Any finite union of disjoint, mutually exterior Jordan curves in the complex plane can be approximated arbitrarily well in the Hausdorff topology by polynomial Julia sets. Furthermore, the proof is… Expand

True trees are dense

- Mathematics
- 2014

We show that any compact, connected set K in the plane can be approximated by the critical points of a polynomial with two critical values. Equivalently, K can be approximated in the Hausdorff metric… Expand

Dipoles and Pixie dust

- Mathematics, Physics
- 2014

Every closed subset of the Riemann sphere can be approximated in the Hausdorff topology by the Julia set of a rational map.

Dynamical dessins are dense

- Mathematics
- 2015

We apply a recent result of the first author to prove the following result: any continuum in the plane can be approximated arbitrarily closely in the Hausdorff topology by the Julia set of a post… Expand

On the (Filled-) Julia sets of Orthogonal polynomials

- Mathematics
- 2016

In this paper we relate the dynamical properties of the sequence of orthonormal polynomials defined by a probability measure $\mu$ with non-polar compact support $S(\mu)$ to the potential theoretic… Expand

Julia Sets of Orthogonal Polynomials

- Mathematics
- 2016

For a probability measure with compact and non-polar support in the complex plane we relate dynamical properties of the associated sequence of orthogonal polynomials {Pn} to properties of the… Expand

Fekete polynomials and shapes of Julia sets

- Mathematics
- Transactions of the American Mathematical Society
- 2019

We prove that a nonempty, proper subset $S$ of the complex plane can be approximated in a strong sense by polynomial filled Julia sets if and only if $S$ is bounded and $\hat{\mathbb{C}} \setminus… Expand

On Lagrange polynomials and the rate of approximation of planar sets by polynomial Julia sets

- Mathematics
- 2018

Abstract We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed in [27] and analyze the rate of approximation. We use slightly modified… Expand

Attractor sets and Julia sets in low dimensions

- Mathematics
- Conformal Geometry and Dynamics of the American Mathematical Society
- 2019

If
X
X
is the attractor set of a conformal IFS (iterated function system) in dimension two or three, we prove that there exists a quasiregular semigroup
G
G
with a Julia set equal… Expand

Quaternion Julia Set Shape Optimization

- Mathematics, Computer Science
- SGP '15
- 2015

This work presents the first 3D algorithm capable of answering the question: what would a Mandelbrot‐like set in the shape of a bunny look like, and shows that it is possible to answer this question by casting it as a shape optimization that discovers novel, highly complex shapes. Expand

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Teichmüller Theory and Applications to Geometry, Topology, and Dynamics

- Mathematics
- 2016

This volume is the second of four volumes devoted to Teichmuller theory and its applications to geometry, topology, and dynamics. The first volume gave an introduction to Teichmuller theory.
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