相关论文: Hyperbolic components of polynomials with a fixed …
In the space of cubic polynomials, Milnor defined a notable curve $\mathcal S_p$, consisting of cubic polynomials with a periodic critical point, whose period is exactly $p$. In this paper, we show that for any integer $p\geq 1$, any…
This article focus on the connected locus of the cubic polynomial slice $Per_1(\lambda)$ with a parabolic fixed point of multiplier $\lambda=e^{2\pi i\frac{p}{q}}$. We first show that any parabolic component, which is a parallel notion of…
We consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself which have degree two or more on each copy. In any space $\p^{S}$ of suitably normalized maps of…
We prove that the boundary of every parabolic component in the cubic polynomial slice $Per_1(1)$ is a Jordan curve by adapting the technique of para-puzzles presented in \cite{Roesch1}. We also give a global description of the connected…
Consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally polynomial maps from a finite union of copies of $\C$ to itself. In the space of suitably normalized maps of this type, the hyperbolic maps form an open set called…
We study the bifurcation loci of quadratic (and unicritical) polynomials and exponential maps. We outline a proof that the exponential bifurcation locus is connected; this is an analog to Douady and Hubbard's celebrated theorem that (the…
We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show ``shrinking of puzzle pieces'' without using specific puzzles.…
A while ago MLC (the conjecture that the Mandelbrot set is locally connected) was proven for quasi-hyperbolic points by Douady and Hubbard, and for boundaries of hyperbolic components by Yoccoz. More recently Yoccoz proved MLC for all at…
The multicorns are the connectedness loci of unicritical antiholomorphic polynomials $\bar{z}^d + c$. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic…
We investigate the discontinuity of codings for the Julia set of a quadratic map. To each parameter ray, we associate a natural coding for Julia sets on the ray. Given a hyperbolic component $H$ of the Mandelbrot set, we consider the…
We exhibit products of Mandelbrot sets in the two-dimensional complex parameter space of cubic polynomials. These products were observed by J. Milnor in computer experiments which inspired Lavaurs' proof of non local-connectivity for the…
The hyperbolic components in the moduli space ${M}_d$ of degree $d\geq2$ rational maps are mysterious and fundamental topological objects. For those in the connectedness locus, they are known to be the finite quotients of the Euclidean…
We establish necessary and sufficient conditions for the realization of mapping schemata as post-critically finite polynomials, or more generally, as post-critically finite polynomial maps from a finite union of copies of the complex…
We studied the parameter plane of the cosine functions with a fixed critical point. The hyperbolic components can be classified into three types: A, C and D. All the hyperbolic components are bounded and simply connected, except for the…
We prove an extension results for the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove…
We prove combinatorial rigidity of infinitely renormalizable unicritical polynomials, P_c :z \mapsto z^d+c, with complex c, under the a priori bounds and a certain "combinatorial condition". This implies the local connectivity of the…
Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number theta, with theta being a given…
In this article, we study the hyperbolic components of McMullen maps. We show that the boundaries of all hyperbolic components are Jordan curves. This settles a problem posed by Devaney. As a consequence, we show that cusps are dense on the…
This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with…
In the moduli space of complex cubic polynomials with a marked critical point, given any p>=1, we prove that the loci formed by polynomials with the marked critical point periodic of period p is an irreducible curve. Thus answering a…