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.     

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.     

A generalized version of Branner-Hubbard conjecture for rational functions

In 1992, Branner and Hubbard raised a conjecture that the Julia set of a polynomial is a Cantor set if and only if each critical component of its filled-in Julia set is not periodic. This conjecture was solved recently. In this paper, we generalize this result to a class of rational functions.     

具有淹没分支的有理函数的性质（英文）

Let R(z) be an NCP map with buried components of degree d = degf ≥ 2 on the complex sphere ■, and HD denotes the Hausdorff dimension. In this paper we prove that if R_n→ R algebraically, and R_n and R topologically conjugate for all n 0, then R_n is an NCP map with buried components for all n 0, and for some C 0,d_H(J(R), J(R_n)) ≤ C(dist(R, R_n))~(1/d),where d_H denotes the Hausdorff distance, and HD(J(R_n)) → HD(J(R)).In this paper we also prove that if the Julia set J(R) of an NCP map R(z) with buried components is locally connected, then any component J_i(R) is either a real-analytic curve or HD(J_i(R)) 1.     

A Note on The Two-Functional Conjecture

Let D={z:|z|<1} and let K(D) denote the set of all functions analytic in D with the usual topology of uniform convergence on compact subsets of D. Let S be the class of function f(z) =z+a_2z~2+…analytic and univalent in D. Then S is a compact subset of K(D). A function f∈S is said to be a support point of S if it maximizes Re{L} over S for some continuous comp-     

A Note on Upper Convex Density

For a self-similar set E satisfying the open set condition,upper convex density is an important concept for the computation of its Hausdorff measure,and it is well known that the set of relative interior points with upper convex density 1 has a full Hausdorff measure.But whether the upper convex densities of E at all the relative interior points are equal to 1？ In other words,whether there exists a relative interior point of E such that the upper convex density of E at this point is less than 1？ In this paper,the authors construct a self-similar set satisfying the open set condition,which has a relative interior point with upper convex density less than 1.Thereby,the above problem is sufficiently answered.     

Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

It is proved that the quadratic system with a weak saddle has at most one limit cycle,andthat if this system has a separatrix cycle passing through the weak saddle,then the stability of theseparatrix cycle is contrary to that of the singular point surrounded by it.     

THE LIMIT SET OF A CONTINUOUS SELF-MAP OF THE TREE

Let f denote a continuous map of a tree T to itself. A point x ∈ T is called a 7-limit point of f if it is both an ω-limit point and an α-limit point. In the present paper, we show that (1) Ω-Γ is countable, (2) A -Γ and P - Γ are either empty or countably infinite, where P denotes the closure of the set of periodic points P.     

和图、整和图与模和图的几个结果

Let N denote the set of positive integers. The sum graph G^＋（S） of a finite subset S belong to N is the graph （S, E） with uv ∈ E if and only if u ＋ v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S belong to N. By using the set Z of all integers instead of N, we obtain the definition of the integral sum graph. A graph G = （V, E） is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2,..., z - 1} so that uv ∈ E if and only if the sum, modulo z, of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill （Dm） is integral sum graph and mod sum graph, and give the sum number of Dm.     

Extending external rays throughout the Julia sets of rational maps

For polynomial maps in the complex plane, the notion of external rays plays an important role in determining the structure of and the dynamics on the Julia set. In this paper we consider an extension of these rays in the case of rational maps of the form Fλ(z) = z ⁿ + λ/z ⁿ where n > 1. As in the case of polynomials, there is an immediate basin of ∞, so we have similar external rays. We show how to extend these rays throughout the Julia set in three specific examples. Our extended rays are simple closed curves in the Riemann sphere that meet the Julia set in a Cantor set of points and also pass through countably many Fatou components. Unlike the external rays, these extended rays cross infinitely many other extended rays in a manner that helps determine the topology of the Julia set.     

Julia directions of meromorphic functions and their derivatives

We prove that if a transcendental meromorphic function has no Julia direction and is bounded on a path to ￥ \infty then there is a common Julia direction for all derivatives. Related statements are obtained under the assumption that f is o(?{ | z | }) o(\sqrt{\mid z \mid}) or O(?{ | z | }) O(\sqrt{\mid z \mid}) on a path to ￥ \infty . Further we disprove a conjecture of Frank and Wang by means of a counterexample.     

On the non-escaping set of <Emphasis Type="Italic">e</Emphasis><Superscript>2<Emphasis Type="Italic">πiθ</Emphasis></Superscript> sin (<Emphasis Type="Italic">z</Emphasis>)

Let 0 < θ < 1 be an irrational number of bounded type. We proved that almost every point in the Julia set of e ^2πiθ sin (z) is an escaping point. Partially supported by National Basic Research Program of China (973 Program) 2007CB814800.     

On the dynamics of the singularly perturbed rational maps

In this paper, we study the dynamics of the family of rational maps fλ,（z） = zn - λ/zm, n ≥2, m ≥ 1,λ ∈ C. We construct an example of buried Sierpinski curve Julia set in this family. We also give an estimate of the location of bifurcation locus of fλ.     

Julia set of λ exp(<Emphasis Type="Italic">z</Emphasis>)/<Emphasis Type="Italic">z</Emphasis> with real parameters λ

In this paper, we investigate the Julia set of the family λ exp(z)/z with real parameters λ. We look for what values of real parameters λ such that the Julia set of λ exp(z)/z does not coincide with the whole plane, and thus gives a complete classification for real parameters, which is similar to Jang’s result of a family of transcendental entire functions. Moreover, We also discuss the shape and size of Fatou sets and Julia sets of λ exp(z)/z with real parameters λ when the Julia sets are not the whole plane.     

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.     

Entire functions with Julia sets of positive measure

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy–Carleman–Ahlfors theorem implies that if the set of all z for which |f(z)| > R has N components for some R > 0, then the order of f is at least N/2. More precisely, we have log log M(r, f) ≥ (N/2) log r − O(1), where M(r, f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Carleman and Tsuji related to the Denjoy–Carleman–Ahlfors theorem.     

Julia sets and periodic kneading sequences

We construct abstract Julia sets homeomorphic to Julia sets for complex polynomials of the form f _c (z) = z ² + c, having an associated periodic kneading sequence of the form [`(a*)]{\overline{\alpha\ast}} which is not a period n-tupling. We show that there is a single simply-defined space of “itineraries” which contains homeomorphic copies of all such Julia sets in a natural combinatorial way, with dynamical properties which are derivable directly from the combinatorics. This also leads to a natural definition of abstract Julia sets even for those kneading sequences which are not realized by any polynomial f _c , with similar dynamical properties.     

Residual Julia set for functions with the Ahlfors' Property

We consider two classes of functions studied by Epstein [A.L. Epstein, Towers of finite type complex analytic maps, Ph.D. thesis, City University of New York, 1993] and by Herring [M.E. Herring, An extension of the Julia–Fatou theory of iteration, Ph.D. thesis, Imperial College, London, 1994], which have the Ahlfors' Property. We prove under some conditions on the Fatou and Julia sets that the singleton buried components are dense in the Julia set for these classes of functions.