Showing papers on "External ray published in 1993"
TL;DR: In this article, an octonionic generalization of the Mandelbrot set is proposed which is sensitive to these transition points, and the transition point at which such effects are observed corresponds to the structural phase transitions previously identified.
Abstract: A closer investigation of octonionic Julia sets of a modified quadratic map reveals a direct connection between the strength of the non-associativity parameter and the existence of extended attracting objects. The ‘transition point’ at which such effects are observed corresponds to the structural phase transitions previously identified, and is shown to be approximately linearly related to the characteristic parameter c . An octonionic generalization of the Mandelbrot set is proposed which is sensitive to these transition points.
16 citations
TL;DR: In this paper, the distribution of hyperbolic components along the real axis is discussed and expressions for the number of Hyperbolic Components, cardioids, and discs associated with N -cycles for the Mandelbrot set are given.
14 citations
TL;DR: In this article, the authors give an abstract description of the equivalence relation approximately, discuss results by Thurston and Lavaurs from the viewpoint of symbolic dynamics and study renormalization in the abstract Mandelbrot set.
Abstract: For pt.I see Ergodic Theory and Related Topics (Lecture Notes in Mathematics), Berlin: Springer p.1 (1992). Under the assumption that the Mandelbrot set is locally connected, its boundary can be considered as a topological factor T/ approximately of the circle T, which is called the abstract Mandelbrot set. The structure of T/ approximately is tightly connected with the angle-doubling map on T. The authors give an abstract description of the equivalence relation approximately , discuss results by Thurston (1985) and Lavaurs (1986) from the viewpoint of symbolic dynamics and study renormalization in the abstract Mandelbrot set.
13 citations
TL;DR: In this article, the authors consider the rational maps given by z 7! jzj2 2z2 + c, for z and c complex and > 12 fixed and real.
Abstract: Introduction 1. Definitions and Elementary Results 2. When Disconnected Filled-in Julia Sets are Cantor Sets 3. Smooth Julia Sets 4. Fixed Points 5. Remarks on the Topology of the Connectedness Locus Acknowledgements References We consider the rational maps given by z 7! jzj2 2z2 + c, for z and c complex and > 12 fixed and real. The case = 1 corresponds to quadratic polynomials: some of the well-known results for this conformal case still hold for near 1, while others break down. Among the differences between the two cases are the possibility, for 6= 1, of periodic attractors that do not attract the critical point, and the fact that for < 1 the Julia set is smooth for an open set of values of c. Numerical evidence suggests that the analogue of the Mandelbrot set for this family is connected, but not locally connected if 6= 1.
12 citations
TL;DR: If an invertible post-critically finite self-similar set is simply connected, it is homeomorphic to the Julia set for some polynomial map as mentioned in this paper, and if the set is connected, then it is Julia-like.
8 citations
TL;DR: In this article, the authors studied how certain loops in the parameter space of quadratic complex polynomials give rise to shift-automorphisms of quotients of the set Σ 2 of sequences on two symbols.
Abstract: In this paper we study how certain loops in the parameter space of quadratic complex polynomials give rise to shift-automorphisms of quotients of the set Σ 2 of sequences on two symbols. The Mandelbrot set M is the set of parameter values for which the Julia set of the corresponding polynomial is connected. Blanchard, Devaney, and Keen have shown that closed loops in the complement of the Mandelbrot set give rise to shift-automorphisms of Σ 2 , i.e., homeomorphisms of Σ 2 that commute with the shift map. We study what happens when the loops are not entirely in the complement of the Mandelbrot set. We consider closed loops that cross the Mandelbrot set at a single main bifurcation point, surrounding a component of M attached to the main cardioid
3 citations
TL;DR: The structure of the interior of Julia sets and of the Mandelbrot set of quadratic maps is explored and two different ways of coloring and probing into these sets are provided.