A note on a quadratic rational map with two Siegel disks

Suppose f（z） is a quadratic rational map with two Siegel disks. If the rotation numbers of the Siegel disks are both of bounded type, the Hausdorff dimension of the Julia set satisfies Dim （J（f））〈2.     

Hausdorff dimension of Julia set of a polynomial with a Siegel disk

Let f(z) = e2πiθz(1 z/d)d,θ∈R\Q be a polynomial. Ifθis an irrational number of bounded type, it is easy to see that f(z) has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension of the Julia set of f(z) satisfies Dim(J(f))<2.     

On conformal measures for infinitely renormalizable quadratic polynomials

We study a conformal measure for an infinitely renormalizable quadratic poly- nomial.We prove that the conformal measure is ergodic if the polynomial is unbranched and has complex bounds.The main technique we use in the proof is the three-dimensional puzzle for an infinitely renormalizable quadratic polynomial.     

Hausdorff dimension of quadratic rational Julia sets

Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.     

Continuity of Julia sets

A sufficient and necessary condition is given for the continuity of Julia sets in the space of all rational maps with degreek>1. Project supported by the National Natural Science Foundation of China (Grant No. 19871002).     

On the Classification of Laminations Associated to Quadratic Polynomials

Given any rational map f, there is a lamination by Riemann surfaces associated to f. Such laminations were constructed, in general, by Lyubich and Minsky. In this article, we classify laminations associated to quadratic polynomials with periodic critical point. In particular, we prove that the topology of such laminations determines the combinatorics of the parameter. We also describe the topology of laminations associated to other types of quadratic polynomials.      

The topology of Julia sets for polynomials

We prove that wandering components of the Julia set of a polynomial are singletons provided each critical point in a wandering Julia component is non-recurrent. This means a conjecture of Branner-Hubbard is true for this kind of polynomials     

Stability of Julia sets for a quadratic random dynamical system

For a sequence (cⁿ) of complex numbers, the quadratic polynomials f_cn:= z² + C_n and the sequence (F_n) of iterates F_n: = f_cn ο ⋯ ο f_c1 are considered. The Fatou set F(C_n) is defined as the set of all such that (Fⁿ) is normal in some neighbourhood of z, while the complement J(C_n) of F(c_n) (in ) is called the Julia set. The aim of this paper is to study the stability of the Julia set J(C_n) in the case where (c_n) is bounded. A problem put forward by Brück is solved.     

On the connectivity of the Julia set of a finitely generated rational semigroup

In this paper we show that the Julia set of a finitely generated rational semigroup is connected if the union of the Julia sets of generators is contained in a subcontinuum of . Under a nonseparating condition, we prove that the Julia set of a finitely generated polynomial semigroup is connected if its postcritical set is bounded.

     

K3 surfaces,Picard numbers and Siegel disks

If a K3 surface admits an automorphism with a Siegel disk, then its Picard number is an even integer between 0 and 18. Conversely, using the method of hypergeometric groups, we are able to construct K3 surface automorphisms with Siegel disks that realize all possible Picard numbers. The constructions involve extensive computer searches for appropriate Salem numbers and computations of algebraic numbers arising from holomorphic Lefschetz-type fixed point formulas and related Grothendieck residues.     

Dynamical convergence of polynomials to the exponential

In this paper we investigate the relationship between the dynamics of the polynomials maps p_d,λ(z)=(1+z/d)^d and the exponential family E_λ(z)=λc^z. We show that the hyperbolic components of the parameter planes for the polynomials converge to those for the exponential family as the degree d tends to infinity. We also show that certain "hairs"in the parameter plane for the exponential are limits of correspondings external rays for the polynomial families. For parameter values on the hairs, the Julia sets for the corresponding exponentials are the entire plane whereas, for polynomial parameters on the external rays, the Julia sets are Cantor sets     

Topological entropy of quadratic polynomials and dimension of sections of the Mandelbrot set

A fundamental theme in holomorphic dynamics is that the local geometry of parameter space (e.g. the Mandelbrot set) near a parameter reflects the geometry of the Julia set, hence ultimately the dynamical properties, of the corresponding dynamical system. We establish a new instance of this phenomenon in terms of entropy.     

Dynamics of Polynomial Automorphisms of C~N

ＤｙｎａｍｉｃｓｏｆＰｏｌｙｎｏｍｉａｌＡｕｔｏｍｏｒｐｈｉｓｍｓｏｆＣ~Ｎ￥ＺｈａｎｇＷｅｎｊｕｎ（ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ，Ｋａｉｆｅｎｇ，Ｐ．Ｒ．Ｃｈｉａｎ，４７５００１）Ａｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｉｓａｓｓｉｇｎｅｄｔｏｄｉｓ?..     

Hardy–Weinberg Theory for Tetraploidy with Mixed Mating

The dynamics of tetraploidy with mixed mating are studied, with fitness of inbreeders as a parameter. We determine the fixed points and the asymptotic behavior of the trajectories in the gametic and zygotic structures associated to this reproductive system. Our methods include elementary commutative algebra and algebraic geometry, and we make pervasive use of the computer program Macaulay2.     

Building blocks for Quadratic Julia sets

We obtain results on the structure of the Julia set of a quadratic polynomial with an irrationally indifferent fixed point in the iterative dynamics of . In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set : there exists a nowhere dense subcontinuum such that , is the union of the impressions of a minimally invariant Cantor set of external rays, contains the critical point, and contains both the Cremer point and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and contains no periodic points. In both cases, the Julia set is the closure of a skeleton which is the increasing union of countably many copies of the building block joined along preimages of copies of a critical continuum containing the critical point. In addition, we prove that if is any polynomial of degree with a Siegel disk which contains no critical point on its boundary, then the Julia set is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.

     

On biaccessible points in the Julia set of a Cremer quadratic polynomial

We prove that the only possible biaccessible points in the Julia set of a Cremer quadratic polynomial are the Cremer fixed point and its preimages. This gives a partial answer to a question posed by C. McMullen on whether such a Julia set can contain any biaccessible point at all.

     

Not all Julia sets are quasi-self-similar

We show that there exist rational functions, whose Julia set fails to be quasi-self-similar.