A frequent problem in holomorphic dynamics is to prove local connectivity of Julia sets and of many points of the Mandelbrot set; local connectivity has many interesting implications. The intention of this paper is to present a new point of view for this problem: we introduce fibers of these sets, and the goal becomes to show that fibers are ``trivial'', i.e. they consist of single points. The idea is to show ``shrinking of puzzle pieces'' without using specific puzzles. This implies local connectivity at these points, but triviality of fibers is a somewhat stronger property than local connectivity. Local connectivity proofs in holomorphic dynamics often actually yield that fibers are trivial, and this extra knowledge is sometimes useful.
Since we believe that fibers may be useful in further situations, we discuss their properties for arbitrary compact connected and full sets in the complex plane. This allows to use them for connected filled-in Julia sets of polynomials, and polynomials of the form $z^d+c$ have the property that the impression of any dynamic ray at a rational angle is a single point. An appendix reviews known topological properties of compact, connected and full sets in the plane.
The definition of fibers grew out of a new brief proof that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. This proof works also for ``Multibrot sets'', which are the higher degree cousins of the Mandelbrot set. These sets are discussed in a self-contained sequel (IMS Preprint 1998/13a). Finally, we relate triviality of fibers to tuning and renormalization in IMS Preprint 1998/13b.
Local stable and unstable sets for random dynamical systems
School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）
In this paper we give a unified proof of the fact that the Julia set of Newton's method applied to a holomorphic function of the complex plane (a polynomial of degree large than $1$ or an entire transcendental function) is connected. The result was recently completed by the authors' previous work, as a consequence of a more general theorem whose proof spreads among many papers, which consider separately a number of particular cases for rational and transcendental maps, and use a variety of techniques. In this note we present a unified, direct and reasonably self-contained proof which works for…
Using computer graphics and visualization algorithms, we extend in this work the results obtained analytically in [1], on the connectivity domains of alternated Julia sets, defined by switching the dynamics of two quadratic Julia sets. As proved in [1], the alternated Julia sets exhibit, as for polynomials of degree greater than two, the disconnectivity property in addition to the known dichotomy property (connectedness and totally disconnectedness) which characterizes the standard Julia sets. Via computer graphics, we unveil these connectivity domains which are four-dimensional fractals. The …
We explore geometric properties of the Mandelbrot set M, and the corresponding Julia sets J_c, near the main cardioid. Namely, we establish that: a) M is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; b) The Julia sets J_c are also locally connected and have positive area; c) M is self-similar near Siegel parameters of constant type. We approach these problems by analyzing the unstable manifold of the pacman renormalization operator constructed in [DLS] as a global transcendental family.
We study tangent sets of strictly self-affine sets in the plane. If a set in this class satisfies the strong separation condition and projects to a line segment for sufficiently many directions, then for each generic point there exists a rotation $\mathcal O$ such that all tangent sets at that point are either of the form $\mathcal O((\mathbb R \times C) \cap B(0,1))$, where $C$ is a closed porous set, or of the form $\mathcal O((\ell \times \{ 0 \}) \cap B(0,1))$, where $\ell$ is an interval.
The theory of complex trees is introduced as a new approach to study a broad class of self-similar sets. Systems of equations encoded by complex trees tip-to-tip equivalence relations are used to obtain one-parameter families of connected self-similar sets $F_\mathcal{A}(z)$. In order to study topological changes of $F_\mathcal{A}(z)$ in regions $\mathcal{R}\subset\mathbb{C}$ where these families are defined, we introduce a new kind of set $\mathcal{M}\subseteq\mathcal{R}$ which extends the usual notion of connectivity locus for a parameter space. Moreover we consider another set $\mathcal{M}_…
Rigidity and non local connectivity of Julia sets of some quadratic polynomials
对于具有重整化周期序列${k_m}$和旋转数序列${t_m=p_m/q_m}, we prove that if $ \limsup k_m^{-1} \log |p_m| >0 $, then the Mandelbrot set is locally connected at $ c $. We prove also that if $ \limsup的无限可重整化二次映射$f_c: z\mapsto z^2+c$| t_{m+1}| ^{1/q_m}<1 $ and $ q_m \to \infty $, then the Julia set of $ f_c $ is not locally connected and the Mandelbrot set is locally connected at $ c$，条件是所有的重正化都是非基元（卫星）。这量化了A的构造。Douady和J. Hubbard提出的一个条件，并削弱了J. Milnor提出的一个条件.附录摘要：我们改进了上述论文的一个主要结果。
