scholarly journals From Cantor to semi-hyperbolic parameters along external rays

2019 ◽  
Vol 372 (11) ◽  
pp. 7959-7992
Author(s):  
Yi-Chiuan Chen ◽  
Tomoki Kawahira
Keyword(s):  
External Rays

Drawing and computing external rays in the multiple-spiral medallions of the Mandelbrot set

2008 ◽  
Vol 32 (5) ◽  
pp. 597-610 ◽  
Author(s):  
M. Romera ◽  
G. Alvarez ◽  
D. Arroyo ◽  
A.B. Orue ◽  
V. Fernandez ◽  
...  
Keyword(s):  
Mandelbrot Set ◽  
External Rays

External rays to periodic points

1996 ◽  
Vol 94 (1) ◽  
pp. 29-57 ◽  
Author(s):  
G. Levin ◽  
F. Przytycki
Keyword(s):  
Periodic Points ◽  
External Rays

scholarly journals Operating with External Arguments of Douady and Hubbard

2007 ◽  
Vol 2007 ◽  
pp. 1-17 ◽  
Author(s):  
G. Pastor ◽  
M. Romera ◽  
G. Alvarez ◽  
J. Nunez ◽  
D. Arroyo ◽  
...  
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Typical Case ◽  
Mandelbrot Set ◽  
Nonlinear Phenomena ◽  
External Rays ◽  
General Method

The external arguments of the external rays theory of Douady and Hubbard is a valuable tool in order to analyze the Mandelbrot set, a typical case of discrete dynamical system used to study nonlinear phenomena. We suggest here a general method for the calculation of the external arguments of external rays landing at the hyperbolic components root points of the Mandelbrot set. Likewise, we present a general method for the calculation of the external arguments of external rays landing at Misiurewicz points.

scholarly journals A landing theorem for entire functions with bounded post-singular sets

2020 ◽  
Vol 30 (6) ◽  
pp. 1465-1530
Author(s):  
Anna Miriam Benini ◽  
Lasse Rempe
Keyword(s):  
Entire Function ◽  
Periodic Point ◽  
Infinite Order ◽  
Entire Functions ◽  
Complex Polynomial ◽  
Landing Point ◽  
External Rays ◽  
Singular Sets ◽  
External Ray ◽  
Polynomial Dynamics

AbstractThe Douady-Hubbard landing theorem for periodic external rays is one of the cornerstones of the study of polynomial dynamics. It states that, for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray. We prove an analogue of this theorem for an entire function f with bounded postsingular set. If f has finite order of growth, then it is known that the escaping set I(f) contains certain curves called periodic hairs; we show that every periodic hair lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic hair. For a postsingularly bounded entire function f of infinite order, such hairs may not exist. Therefore we introduce certain dynamically natural connected subsets of I(f), called dreadlocks. We show that every periodic dreadlock lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic periodic point is the landing point of at least one periodic dreadlock. More generally, we prove that every point of a hyperbolic set is the landing point of a dreadlock.

Extending external rays throughout the Julia sets of rational maps

2010 ◽  
Vol 7 (1) ◽  
pp. 223-240 ◽  
Author(s):  
Figen Çilingir ◽  
Robert L. Devaney ◽  
Elizabeth D. Russell
Keyword(s):  
Julia Sets ◽  
Rational Maps ◽  
External Rays

scholarly journals Non-Computable Impressions of Computable External Rays of Quadratic Polynomials

2014 ◽  
Vol 335 (2) ◽  
pp. 739-757 ◽  
Author(s):  
Ilia Binder ◽  
Cristobal Rojas ◽  
Michael Yampolsky
Keyword(s):  
Quadratic Polynomials ◽  
External Rays

scholarly journals Tessellation and Lyubich–Minsky laminations associated with quadratic maps, I: pinching semiconjugacies

2009 ◽  
Vol 29 (2) ◽  
pp. 579-612
Author(s):  
TOMOKI KAWAHIRA
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Quadratic Maps ◽  
Quasiconformal Deformation ◽  
External Rays

AbstractWe construct tessellations of the filled Julia sets of hyperbolic and parabolic quadratic maps. The dynamics inside the Julia sets are then organized by tiles which play the role of the external rays outside. We also construct continuous families of pinching semiconjugacies associated with hyperbolic-to-parabolic degenerations without using quasiconformal deformation. Instead, we achieve this via tessellation and investigation of the hyperbolic-to-parabolic degeneration of linearizing coordinates inside the Julia set.

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