Non-computable Julia sets

We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets.

     

Some cubic Julia sets

We figure out geometric properties of the Julia setJ _a of cubic complex polynomialC _a(z) =z ³ +az(a ∈ ?) and the smallest ellipse which surroundsJ _a.     

Alternate superior Julia sets

Alternate Julia sets have been studied in Picard iterative procedures. The purpose of this paper is to study the quadratic and cubic maps using superior iterates to obtain Julia sets with different alternate structures. Analytically, graphically and computationally it has been shown that alternate superior Julia sets can be connected, disconnected and totally disconnected, and also fattier than the corresponding alternate Julia sets. A few examples have been studied by applying different type of alternate structures.     

Translation invariant Julia sets

We show that if the Julia set of a rational function is invariant under translation by one and infinity is a periodic or preperiodic point for , then must either be a line or the Riemann sphere.

     

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).     

Topological complexity of Julia sets

The topological structures of the Julia sets of rational and entire functions have been investigated and the complexity of the Julia sets of rational functions has been described. For entire functions, it is proved that the dynamics on the Fatou sets will influence the topological complexity of Julia sets.     

On Julia sets concerning phase transitions

The sets of the points corresponding to the phase transitions of the Potts model on the diamond hierarchical lattice for antiferromagnetic coupling are studied. These sets are the Julia sets of a family of rational mappings. It is shown that they may be disconnected sets. Furthermore, the topological structures of these sets are described completely.     

Renormalization group method and Julia sets

The renormalization group (RG) method has been used successfully in treating a variety of phase change and critical-point problems (Wilson KG, Kogut J. Phys Rev C 1974;12:75; Wilson KG. Rev Mod Phys 1975;773; Wilson KG. Phys Rev B 1971;3174). A relatively simple system is considered at the smallest scale; the problem is then renormalized in order to utilize the same system at next larger scale. The process is repeated at larger and larger scales. In the following we consider a model for the flow of a fluid through a porous-medium. The RG transformations for the flow of a fluid through a porous-medium in two and three dimensions are derived and generalized to the complex plane, and the types of the corresponding Julia sets are found and generated. Also, the RG transformation for Ising model on a square lattice is derived and the corresponding Julia set is found.     

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.

     

Not all Julia sets are quasi-self-similar

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

     

Julia sets of permutable Holomorphic maps

Let f and g be two permutable transcendental holomorphic maps in the plane. We shall discuss the dynamical properties of f, g and f o g and prove, among other things, that if either f has no wandering domains or f is of bounded type, then the Julia sets of f and f(g) coincide. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Julia sets for certain rational functions

Julia sets or F sets, have been of considerable interest in current research. In this paper we find a new characterization of the Julia set for certain rational functions and find bounds for its Hausdorff dimension.     

Locally connected models for Julia sets

Let P be a polynomial with a connected Julia set J. We use continuum theory to show that it admits a finest monotone map φ onto a locally connected continuumJP_∼, i.e. a monotone map φ:J→JP_∼ such that for any other monotone map ψ:J→J^′ there exists a monotone map h with ψ=h°φ. Then we extend φ onto the complex plane C (keeping the same notation) and show that φ monotonically semiconjugates PC_| to a topological polynomialg:C→C. If P does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi's fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a characterization and a useful sufficient condition for the map φ not to collapse all of J into a point.     

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.     

牛顿变换Julia集的对称性

主要讨论多项式的牛顿变换Julia集的对称性问题．利用复动力系统理论,证明了多项式P（z）的Julia集的对称群是其牛顿变换Np（z）的Julia集的对称群的子群．获得了Julia集为一水平直线的充分必要条件．     

Julia sets that are full of holes

Conclusion  Many of the most fundamental properties, such as measure and dimension, remain unknown for most Julia sets. Although there are Julia sets that are the whole Riemann sphere and so have dimension two and positive measure, no other Julia sets of measure bigger than zero have been found. Shishikura’s surprising result (1998) shows that there are other Julia sets of dimension 2, which makes it appear possible that there are other Julia sets of positive measure. Proving that a Julia set is full of holes, or porous, provides a bound on the upper box dimension, but this has so far been possible only for special classes of Julia sets. Mean porosity and mean e-porosity, both found in Koskela and Rohde (1997), provide better dimension bounds; nonuniform porosity (Roth 2006) implies measure zero, but is not known to provide dimension bounds. These notions can be used in some cases when it is not possible to prove porosity. In the end, we do not know in general which Julia sets are porous and which are not. In fact, forJ _R, little is known about its dimension or measure. There is much left to explore.     

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