Abstract:
In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a polynomial is always computable.

Abstract:
We give an introduction to buried points in Julia sets and a list of questions about buried points, written to encourage aficionados of topology and dynamics to work on these questions.

Abstract:
本文研究了NCP映射的Julia集为Jordan曲线的问题.利用网格和共形迭代函数系统的方法，获得了Julia集在那种情况下为Jordan曲线的一般结果，推广了有理函数的Julia集为Jordan曲线的复解析动力系统方面的结果. In this paper, we consider the Julia sets of NCP maps as Jordan curves. By the way of net and conformal iterated function system, we obtain the general result of the Julia set in which case as the Jordan curves, which generalizes the results of the complex analytic dynamical systems of Julia sets of rational functions as Jordan curves

Abstract:
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.

Abstract:
Let $f(z) = z^2 + c$ be a quadratic polynomial, with c in the Mandelbrot set. Assume further that both fixed points of f are repelling, and that f is not renormalizable. Then we prove that the Julia set J of f is holomorphically removable in the sense that every homeomorphism of the complex plane to itself that is conformal off of J is in fact conformal on the entire complex plane. As a corollary, we deduce that the Mandelbrot Set is locally connected at such c.

Abstract:
We show that the Julia set of quadratic maps with parameters in hyperbolic components of the Mandelbrot set is given by a transseries formula, rapidly convergent at any repelling periodic point. Up to conformal transformations, we obtain $J$ from a smoother curve of lower Hausdorff dimension, by replacing pieces of the more regular curve by increasingly rescaled elementary "bricks" obtained from the transseries expression. Self-similarity of $J$, up to conformal transformation, is manifest in the formulas. The Hausdorff dimension of $J$ is estimated by the transseries formula. The analysis extends to polynomial maps.

Abstract:
We extend results by Barnsley et al. about orthogonal polynomials on Julia sets to the case of generalized Julia sets. The equilibrium measure is considered. In addition, we discuss optimal smoothness of Green functions and Parreau-Widom criterion for a special family of real generalized Julia sets.

Abstract:
We continue the study of constructing invariant Laplacians on Julia sets, and studying properties of their spectra. In this paper we focus on two types of examples: 1) Julia sets of cubic polynomials $z^3 + c$ with a single critical point; 2) formal matings of quadratic Julia sets. The general scheme introduced in earlier papers in this series involves realizing the Julia set as a circle with identifications, and attempting to obtain the Laplacian as a renormalized limit of graph Laplacians on graphs derived form the circle with identifications model. In the case of cubic Julia sets the details follows the pattern established for quadratic Julia sets, but for matings the details are quite challenging, and we have only been completely successful for one example. Once we have constructed the Laplacian, we are able to use numerical methods to approximate the eigenvalues and eigenfunctions. One striking observation from the data is that for the cubic Julia sets the multiplicities of all eigenspaces (except for the trivial eigenspace of constants) are even numbers. Nothing like this is valid for the quadratic julia sets studied earlier. We are able to explain this, based on the fact that three is an odd number, and more precisely because the dihedral-3 symmetry group has only two distinct one-dimensional irreducible representations.