Filled Julia Set Research Articles

Disconnected Julia sets of quartic polynomials and a new topology of the symbol space

For a certain quartic polynomial, there exists a homeomorphism between the set of all components of the filled-in Julia set with the Hausdorff metric and some subset of the corresponding symbol space with the ordinary metric. But these sets are not compact with respect to each metric. We introduce a new topology with respect to which these sets are compact.

On a certain kind of polynomials of degree 4 with disconnected Julia set

For a polynomial of degree at least two, the Julia set and thefilled-in Julia set are either connected or have uncountably manycomponents. In the case that the Julia set of a polynomial of degree 4is neither connected nor totally disconnected, there exists ahomeomorphism between the set of all components of the filled-inJulia set and some subset of the corresponding symbol space.Furthermore the polynomial is topologically conjugate to the shiftmap via the homeomorphism. Moreover there exists ahomeomorphism between the Julia sets of the polynomial and that of acertain polynomial semigroup.

Visual presentation of dynamic systems with hyperbolic planar symmetry

Hyperbolic symmetric mappings defined on hyperbolic tilings are investigated. Ljapunov exponents of the dynamic systems are computed with the Euclidean distance. The parameter combinations with great impact on the characteristics of the dynamic systems were chosen as the window coordinates for construction of generalized Mandelbrot sets. The accelerated direct search algorithm is used to search for the set of the critical points in the fundamental region. The parameter space is separated into chaotic, periodic and mixed regions by the Ljapunov exponents of the critical points. The generalized Mandelbrot sets (M-set), which are the cross-sections of the parameter space, were constructed. Three different types of hyperbolic symmetry patterns, which are chaotic attractors, filled-in Julia sets and mixed images composed of an attractor and a filled-in Julia set from the same set of parameters, were created by using parameters from this kind of M-sets.

Preperiodic points of polynomials over global fields

Given a global field K and a polynomial defined over K of degree at least two, Morton and Silverman conjectured in 1994 that the number of K-rational pre-periodic points of is bounded in terms of only the degree of K and the degree of . In 1997, for quadratic polynomials over K = ℚ, Call and Goldstine proved a bound which was exponential in s, the number of primes of bad reduction of . By careful analysis of the filled Julia sets at each prime, we present an improved bound on the order of s log s. Our bound applies to polynomials of any degree (at least two) over any global field K.

Transfinite diameter and the resultant

We prove a formula for the Fekete-Leja transfinite diameter of the pullback of a set E in C^N by a regular polynomial map F, expressing it in terms of the resultant of the leading part of F and the transfinite diameter of E. We also establish the nonarchimedean analogue of this formula. A key step in the proof is a formula for the transfinite diameter of the filled Julia set of F.

Puzzle geometry and rigidity: The Fibonacci cycle is hyperbolic

We describe a new and robust method to prove rigidity results in complex dynamics. The new ingredient is the geometry of the critical puzzle pieces: under control of geometry and “complex bounds”, two generalized polynomial-like maps which admit a topological conjugacy, quasiconformal outside the filled-in Julia set, are indeed quasiconformally conjugate. The proof uses a new abstract removability-type result for quasiconformal maps, following ideas of Heinonen and Koskela and of Kallunki and Koskela, optimized for applications in complex dynamics. We prove, as the first application of this new method, that, for even criticalities distinct from two, the period two cycle of the Fibonacci renormalization operator is hyperbolic with 1 1 -dimensional unstable manifold.

Filled Julia Sets with Empty Interior Are Computable

Filled Julia Sets with Empty Interior Are Computable

Generalized Binet dynamics

The Binet formula provides a mechanism for the Fibonacci numbers to be viewed as a function of a complex variable. The Binet formula may be generalized by using other bases and multiplicative parameters that also give functions of a complex variable. Thus, filled-in Julia sets that exhibit escape time may be constructed. Moreover, these functions have computable critical points and hence we can create escape time images of the critical point based upon the underlying multiplicative parameter. Like the classic Mandelbrot set, these parameter space images provide an atlas of Julia sets.

Homeomorphisms from Julia sets into connectedness loci

We construct a homeomorphism from the filled Julia set of a given polynomial into the connectedness locus of the family of all polynomials of higher degree by using intertwining surgery. A polynomial in the image of such a homeomorphism has a renormalization of ‘capture’ type and the inverse map is given by the ‘marking’ of the captured critical orbit.

Critical points and dynamic systems with planar hexagonal symmetry

Abstract In this investigation, we detect and utilize critical points of functions with hexagonal symmetry in order to study their dynamics. The asymmetric unit in a parallelogram lattice is chosen as the initial searching region for a critical point set in a dynamic plane. The accelerated direct search algorithm is used within the parallelogram lattice to search for the critical points. Parameter space is separated into regions (chaotic, periodic or mixed) by the Ljapunov exponents of the critical points. Then the generalized Mandelbrot set (M-set), which is a cross-section of the parameter space, is constructed. Many chaotic attractors and filled-in Julia sets can be generated by using parameters from this kind of M-sets.

Strong Analyticity of Partly Filled-in Composite Julia Sets

It is shown that a composite Julia set generated by an infinite array of polynomial mappings is strongly analytic when regarded as a multifunction of the generating maps. An example of such a multifunction, the values of which have Holder Continuity Property, is constructed.

ON A BICOMPLEX DISTANCE ESTIMATION FOR THE TETRABROT

In this article, we present some distance estimation formulas that can be used to ray traced slices of the bicomplex Mandelbrot set and the bicomplex filled-Julia sets in dimension three. We also present a simple method to explore and infinitely approach these 3D fractals. Because of its rich fractal structure and symmetry, we emphasize our work on the generalized Mandelbrot set for bicomplex numbers in dimension three: the Tetrabrot.

DYNAMICS OF A FAMILY OF QUADRATIC MAPS IN THE QUATERNION SPACE

The dynamics of a family of quadratic maps in the quaternion space is investigated. In particular, connectivity of the filled-in Julia sets is completely determined. It is shown that the connectedness locus of this family is not equal to what we call the quaternionic Mandelbrot set. Hyperbolic components will also be completely characterized.

Canonical heights, transfinite diameters, and polynomial dynamics

Let φ(z) be a polynomial of degree at least 2 with coefficients in a number field K. Iterating φ gives rise to a dynamical system and a corresponding canonical height function ĥφ, as defined by Call and Silverman. We prove a simple product formula relating the transfinite diameters of the filled Julia sets of φ over various completions of K, and we apply this formula to give a generalization of Bilu’s equidistribution theorem for sequences of points whose canonical heights tend to zero.

QUADRATIC DYNAMICS IN MATRIX RINGS: TALES OF TERNARY NUMBER SYSTEMS

We describe the quadratic dynamics in certain three-component number systems, which like the complex numbers, can be expressed as rings of real matrices. This description is accomplished using the properties of the real quadratic family and its various first- and second-order phase and parameter derivatives. We demonstrate that the fundamental dichotomy of defining the Mandelbrot set either in terms of filled Julia sets or in terms of the orbit of the origin extends to these ternary number systems.

Views of Fibonacci dynamics

The Binet formula gives a natural way for Fibonacci numbers to be viewed as a function of a complex variable. We experimentally study the complex dynamics of the Fibonacci numbers viewed in that manner. Attracting and repelling fixed points are related to the filled Julia set and to regions of escape time images with fascinating behavior.

THE IMMEDIATE BASIN OF ATTRACTION OF INFINITY FOR POLYNOMIAL SEMIGROUPS OF FINITE TYPE

As a generalization of the dynamics of a single polynomial, Hinkkanen and Martin studied the dynamics of polynomial semigroups of finite type. Among the issues addressed were possible generalizations of the filled-in Julia set of a single polynomial and the basin of attraction for infinity. In the paper, these concepts are further refined, and the connections between the dynamics of a single function and the dynamics of semigroups are strengthened.

Pasting Together Julia Sets: A Worked Out Example of Mating

The operation of “mating” two suitable complex polynomial maps f l and f 2 constructs a new dynamical system by carefully pasting together the boundaries of their filled Julia sets so as to obtain a copy of the Riemann sphere, together with a rational map f l f 2 from this sphere to itself. This construction is particularly hard to visualize when the filled Julia sets K(fi) are dendrites, with no interior. This note will work out an explicit example of this type, with effectively computable maps from K(f l) and K(f 2) onto the Riemann sphere.

Composite Julia sets generated by infinite polynomial arrays

A definition of a generalized filled-in Julia set generated by an infinite array of proper polynomial mappings in C N is introduced. It is shown that such Julia sets depend analytically on the defining polynomial mappings.

Hecke Groups, Polynomial Maps and Matings

We show that there is a topological conjugacy between the action of the Hecke group Gn on the completed positive real axis and the action of the shift on base n-1 expressions of real numbers in the unit interval. This conjugacy is shown to occur in holomorphic dynamical systems: we construct (n-1:n-1) holomorphic correspondences (multivalued self-maps of the Riemann sphere) which are matings between Gn acting on the complex upper half plane, and polynomials of degree n-1 acting on their filled Julia sets. Certain of these correspondences split into unions of (2:2) and (1:1) correspondences: we present combinatorial descriptions of limit sets and the connectivity locus in the case n=4.