Showing papers on "External ray published in 2001"

On limiting directions of julia sets

Abstract: We deal with the iteration of transcendental entire functions, and prove some properties on the Julia sets.

On supports of dynamical laminations and biaccessible points in polynomial Julia sets

Abstract: We use Beurling estimates and Zdunik’s theorem to prove that the support of a lamination of the circle corresponding to a connected polynomial Julia set has zero length, unless f is conjugate to a Chebyshev polynomial. Equivalently, except for the Chebyshev case, the biaccessible points in the connected polynomial Julia set have zero harmonic measure. A connected, locally connected, full, compact subset K of the complex plane C can be topologically described by a lamination of a circle, which tells how to pinch the circle to obtain K (see, e.g., [Dou]). A lamination is an equivalence relation on the unit circle T, identifying points ζ and ζ ′ if they are mapped to one point in K by the Riemann uniformization map of the complement of K. To obtain a topological model of the compact set K, we glue together points of the unit circle, belonging to one equivalence class. The support of a lamination is defined as the union of all non-trivial (containing 2 or more points) equivalence classes, i.e. it includes those points which are identified with some other points. The laminations so defined are topologically fully characterized (among all equivalence relations on T, see [Dou]) by the following properties: (1) the graph {(ζ, ζ ) : ζ ∼ ζ ′} is a closed set in T× T, (2) the convex hulls of different equivalence classes are disjoint, (3) each equivalence class is totally disconnected. There are also analytical properties (e.g. the logarithmic capacity of each equivalence class is zero) which are not fully understood, and it is a difficult open question how to characterize laminations analytically among all equivalence relations on T. It also makes sense to consider laminations corresponding to not necessarily locally connected compacta, but those lamina2000 Mathematics Subject Classification: Primary 37F20; Secondary 30C85, 30D05.

Extensions of homeomorphisms between limbs of the Mandelbrot set

Abstract: Using holomorphic surgery techniques, we construct a homeomorphism between a neighborhood of any limb without root point of the Mandelbrot set and a neighborhood of any other of equal denominator, in such a way that the limbs are mapped to each other. On the limbs, the homeomorphism coincides with that constructed in “Homeomorphisms between limbs of the Mandelbrot set” (J. Geom. Anal. 9 (1999), 327–390) which proves – without assuming local connectivity of the Mandelbrot set – that these maps are compatible with the embedding of the limbs in the plane. Outside the limbs, the constructed extension is quasi-conformal.

Julia and mandelbrot sets of chebyshev families

Abstract: We present one-parameter families of rational functions of arbitrary degree d which are globally generalized polynomial-like of degree d and roughly speaking locally quadratic-like everywhere, where the parameter appears not only as a purely multiplicative factor but also in a more complicated nonlinear way. The connectedness locus of these families contains homeomorphic copies of the Mandelbrot set. Main emphasis is put on the explicit construction (and not as usual on the existence only) of the sets on which generalized polynomial-likeness and quadratic-likeness are given as well as on the explicit description of the regions where the homeomorphic copies of the Mandelbrot set are located.

Some investigations on the dynamics of the family λz exp(z)

Abstract: The paper examines some properties of the dynamics of a family of transcendental entire functions, We prove that for any integer p there exist functions in the family with each having attractive p;-cycle. Moreover, the components of the bifurcation diagram are simply-connected. We characterize the Fatou sets of functions of the family having attractive fixed noints end show that these functions are structural stable For the Julia set we prove that it is a null set for certain parameters and that the Hausdonf dimension of the Julia set is 2 for the entire family.

Hausdorff dimension 2 for Julia sets of quadratic polynomials

Abstract: We consider families of quadratic polynomials which admit parameterisations in a neighbourhood of the boundary of the Mandelbrot set. We show how to find parameters such that the associated Julia sets are of Hausdorff dimension 2.

Non-linear Fractal Interpolating Functions

Abstract: We consider two non-linear generalizations of fractal interpolating functions generated from iterated function systems The first corresponds to fitting data using a Kth-order polynomial, while the second relates to the freedom of adding certain arbitrary functions An escape-time algorithm that can be used for such systems to generate fractal images like those associated with Julia or Mandelbrot sets is also described

From Newton's method to exotic basins Part II: Bifurcation of the Mandelbrot-like sets

Abstract: This is a continuation of the work (Ba) dealing with the family of all cubic rational maps with two supersinks. We prove the existence of the following parabolic bifurcation of Mandelbrot-like sets in the parameter space of this family. Starting from a Mandelbrot-like set in cubic Newton maps and changing parameters in a continuous way, we construct a path of Mandelbrot-like sets ending in the family of parabolic maps with a xed point of multiplier 1. Then it bifurcates into two paths of Mandelbrot-like sets, contained respectively in the set of maps with exotic or non-exotic basins. The non-exotic path ends at a Mandelbrot-like set in cubic polynomials.