The landing of parameter rays at non-recurrent critical portraits

Orbit Portraits of Unicritical Anti-polynomials

Orbit portraits were introduced by Milnor as a combinatorial tool to describe the patterns of all periodic dynamical rays landing on a periodic cycle of a quadratic polynomial. This encodes information about the dynamics and the parameter spaces of these maps. We carry out a similar analysis for unicritical anti-polynomials and give an explicit description of the orbit portraits that can occur for such maps in terms of their characteristic angles, which turns out to be rather restricted when compared with the holomorphic case. Finally, we prove a realization theorem for these combinatorial obj…

具有周期临界轨道的三次多项式映射，第三部分：镶嵌与轨道肖像

Cubic Polynomial Maps with Periodic Critical Orbit, Part III: Tessellations and Orbit Portraits

本文研究三次多项式映射参数空间中周期临界轨道的动力学变化，通过逃逸区域、参数射线落点及周期q镶嵌刻画参数射线与动态射线的关系，并证明落点必为抛物映射或Misiurewicz映射。

Non-recurrent parameter rays of the Mandelbrot set

In this paper, we prove that any parameter ray at a non-recurrent angle $θ$ lands at a non-recurrent parameter $c$ with $θ$ a characteristic angle of $f_c$; and conversely, every non-recurrent parameter $c$ is the landing point of one or two parameter rays at non-recurrent angles, and these angles are exactly the characteristic angles of $f_c$.

Rational Parameter Rays of The Multibrot Sets

We prove a structure theorem for the multibrot sets, which are the higher degree analogues of the Mandelbrot set, and give a complete picture of the landing behavior of the rational parameter rays and the bifurcation phenomenon. Our proof is inspired by previous works of Schleicher and Milnor on the combinatorics of the Mandelbrot set; in particular, we make essential use of combinatorial tools such as orbit portraits and kneading sequences. However, we avoid the standard global counting arguments in our proof and replace them by local analytic arguments to show that the parabolic and the Misi…

An Extended Fatou-Shishikura inequality and wandering branch continua for polynomials

Let $P$ be a polynomial of degree $d$ with Julia set $J_P$. Let $\widetilde N$ be the number of non-repelling cycles of $P$. By the famous Fatou-Shishikura inequality $\widetilde N\le d-1$. The goal of the paper is to improve this bound. The new count includes \emph{wandering collections of non-precritical branch continua}, i.e., collections of continua or points $Q_i\subset J_P$ \emph{all} of whose images are pairwise disjoint, contain no critical points, and contain the limit sets of $\mathrm{eval}(Q_i)\ge 3$ external rays. Also, we relate individual cycles, which are either non-repelling or…