Jan Kiwi
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 repelling and a given rational lamination. At the same time we derive some consequences that this study has regarding the topology of Julia sets.
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Based on the distortion theory developed by Cui--Tan \cite{CT15}, we prove the landing of every parameter ray at critical portraits coming from non-recurrent polynomials, thereby generalizing a result of Kiwi \cite[Corollary]{Ki05}
A classification of bicritical dynamic portraits
本文针对次数 $d\geq2$ 且至少含4个后临界点的抽象双临界动力肖像，研究其可仅由有理Thurston映射实现的分类问题。
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。
It is well known that the dynamical behavior of a rational map $f:\widehat{\mathbb C}\to \widehat{\mathbb C}$ is governed by the forward orbits of the critical points of $f$. The map $f$ is said to be postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle of $f$. We encode the orbits of the critical points of $f$ with a finite directed graph called a ramification portrait. In this article, we study which graphs arise as ramification portraits. We prove that every abstract polynomial portrait is real…
Let $K$ be an algebraically closed field of characteristic zero, and let $\mathcal{K} := K(t)$ be the rational function field over $K$. For each $d \ge 2$, we consider the unicritical polynomial $f_d(z) := z^d + t \in \mathcal{K}[z]$, and we ask the following question: If we fix $α\in \mathcal{K}$ and integers $M \ge 0$, $N \ge 1$, and $d \ge 2$, does there exist a place $\mathfrak{p} \in \operatorname{Spec} K[t]$ such that, modulo $\mathfrak{p}$, the point $α$ enters into an $N$-cycle after precisely $M$ steps under iteration by $f_d$? We answer this question completely, concluding that the a…
Portraits of quadratic rational maps with a small critical cycle
本文研究具有小周期临界环的二次有理映射的前周期点图，给出$n=4$时的猜想性完整列表，并加强$n=2,3$的已有结果，同时探讨图的线性共轭类表示性问题。
On the Canonical Height Gap for Polynomial Maps and Portraits of Preperiodic Points of Polynomials with Height $0$
研究\(\mathbb{P}^{1}\)上多项式映射的典范高度与朴素高度差，建立与次数\(d\)无关的精确界，以此确定\(\mathbb{Q}\)上高度为0的多项式映射的有理预周期点集并分类其画像。
We extend the definition of an orbit portrait to the context of non-autonomous iteration, both for the combinatorial version involving collections of angles and for the dynamic version involving external rays where combinatorial portraits can be realized by the dynamics associated with sequences of polynomials with suitably uniformly bounded degrees and coefficients. We show that, in the case of sequences of polynomials of constant degree, the portraits which arise are eventually periodic which is somewhat similar to the classical theory of polynomial iteration. However, if the degrees of the …
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…
In this paper, we use the Carathéodory Convergence Theory to prove a landing theorem of rays in hyperbolic components with rational arguments. Although the proof is done in the setting of a family of entire transcendental maps with two singular values, the method can be generalized to many other one parameter families, with some modifications.
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…
Let $K$ be a function field over an algebraically closed field $k$ of characteristic $0$, let $φ\in K(z)$ be a rational function of degree at least equal to $2$ for which there is no point at which $φ$ is totally ramified, and let $α\in K$. We show that for all but finitely many pairs $(m,n)\in \mathbb{Z}_{\ge 0}\times \mathbb{N}$ there exists a place $\mathfrak{p}$ of $K$ such that the point $α$ has preperiod $m$ and minimum period $n$ under the action of $φ$. This answers a conjecture made by Ingram-Silverman and Faber-Granville. We prove a similar result, under suitable modification, also w…
In this paper, we consider a one-parameter family of degree $d\ge 2$ rational maps with an automorphism group containing the cyclic group of order $d$. We construct a polynomial whose roots correspond to parameter values for which the corresponding map is post-critically finite with a certain dynamical portrait. Then we prove that the polynomial is irreducible in certain cases.
We give a new proof that all external rays of the Mandelbrot set at rational angles land, and of the relation between the external angle of such a ray and the dynamics at the landing point. Our proof is different from the original one, given by Douady and Hubbard and refined by Lavaurs, in several ways: it replaces analytic arguments by combinatorial ones; it does not use complex analytic dependence of the polynomials with respect to parameters and can thus be made to apply for non-complex analytic parameter spaces; this proof is also technically simpler. Finally, we derive several corollaries…
On the number of quadratic polynomials with a given portrait
本研究在Morton-Silverman猜想下，针对数域上的二次多项式，给出了按高度计数具有指定有理预周期点画像的参数c的渐近公式，并将结果推广至二次扩域情形。
It is well known that every rational parameter ray of the Mandelbrot set lands at a single parameter. We study the rational parameter rays of the multicorn $\mathcal{M}_d^*$, the connectedness locus of unicritical antiholomorphic polynomials of degree $d$, and give a complete description of their accumulation properties. One of the principal results is that the parameter rays accumulating on the boundaries of odd period (except period $1$) hyperbolic components of the multicorns do not land, but accumulate on arcs of positive length consisting of parabolic parameters. We also show the existenc…
Dynamical approximation of postsingularly finite exponentials
本文构造了一列后临界有限单临界多项式局部一致逼近任意后奇异有限指数函数，并揭示了参数射线角度与指数参数平面外部地址之间的关系。
This paper studies the behavior under iteration of the maps T_{ab}(x,y) = (F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and bx if x<0. The orbits under iteration correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This family of piecewise-linear maps has the parameter space (a,b)\in R^2. These maps are area-preserving homeomorphisms of R^2 that map rays from the origin into rays from the origin. The action on rays defines a map S_{ab} of the circle, which has a well-defined rotation number. This paper characterizes …
Continuity of Julia sets and invariant rays
本文研究了有理映射在抛物线扰动下的Julia集连续性与不变射线的关系，通过抛物线冲击理论建立了等价刻画。
Douady and Hubbard introduced the operation of mating of polynomials. This identifies two filled Julia sets and the dynamics on them via external rays. In many cases one obtains a rational map. Here the opposite question is tackled. Namely we ask when a given (postcritically finite) rational map $f$ arises as a mating. A sufficient condition when this is possible is given. If this condition is satisfied, we present a simple explicit algorithm to unmate the rational map. This means we decompose $f$ into polynomials, that when mated yield $f$. Several examples of unmatings are presented.
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