Sierpiński and non-Sierpiński curve Julia sets in families of rational maps

2008 ◽  
Vol 78 (2) ◽  
pp. 290-304 ◽  
Author(s):  
Norbert Steinmetz
Keyword(s):  
Julia Sets ◽  
Rational Maps ◽  
Sierpiński Curve ◽  
Sierpinski Curve

Cantor sets of circles of Sierpiński curve Julia sets

2007 ◽  
Vol 27 (5) ◽  
pp. 1525-1539 ◽  
Author(s):  
ROBERT L. DEVANEY
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Cantor Set ◽  
Rational Maps ◽  
Closed Curves ◽  
Open Set ◽  
Dynamical Plane ◽  
Sierpiński Curve ◽  
Sierpinski Curve ◽  
Subshifts Of Finite Type

AbstractOur goal in this paper is to give an example of a one-parameter family of rational maps for which, in the parameter plane, there is a Cantor set of simple closed curves consisting of parameters for which the corresponding Julia set is a Sierpiński curve. Hence, the Julia sets for each of these parameters are homeomorphic. However, each of the maps in this set is dynamically distinct from (i.e. not topologically conjugate to) any other map in this set (with only finitely many exceptions). We also show that, in the dynamical plane for any map drawn from a large open set in the connectedness locus in this family, there is a Cantor set of invariant simple closed curves on which the map is conjugate to the product of certain subshifts of finite type with the maps $z \mapsto \pm z^n$ on the unit circle.

scholarly journals Sierpinski curve Julia sets for quadratic rational maps

2014 ◽  
Vol 39 ◽  
pp. 3-22 ◽  
Author(s):  
Robert L. Devaney ◽  
Núria Fagella ◽  
Antonio Garijo ◽  
Xavier Jarque
Keyword(s):  
Julia Sets ◽  
Rational Maps ◽  
Sierpiński Curve ◽  
Sierpinski Curve

scholarly journals Non-landing hairs in Sierpiński curve Julia sets of transcendental entire maps

2011 ◽  
Vol 214 (2) ◽  
pp. 135-160 ◽  
Author(s):  
Antonio Garijo ◽  
Xavier Jarque ◽  
Mónica Moreno Rocha
Keyword(s):  
Julia Sets ◽  
Sierpiński Curve ◽  
Sierpinski Curve

Sierpinski-curve Julia sets and singular perturbations of complex polynomials

2005 ◽  
Vol 25 (4) ◽  
pp. 1047-1055 ◽  
Author(s):  
PAUL BLANCHARD ◽  
ROBERT L. DEVANEY ◽  
DANIEL M. LOOK ◽  
PRADIPTA SEAL ◽  
YAKOV SHAPIRO
Keyword(s):  
Singular Perturbations ◽  
Julia Sets ◽  
Complex Polynomials ◽  
Sierpiński Curve ◽  
Sierpinski Curve

scholarly journals Dynamic classification of escape time Sierpiński curve Julia sets

2009 ◽  
Vol 202 (2) ◽  
pp. 181-198 ◽  
Author(s):  
Robert L. Devaney ◽  
Kevin M. Pilgrim
Keyword(s):  
Julia Sets ◽  
Escape Time ◽  
Sierpiński Curve ◽  
Sierpinski Curve

scholarly journals Buried Sierpinski curve Julia sets

2005 ◽  
Vol 13 (4) ◽  
pp. 1035-1046 ◽  
Author(s):  
Robert L. Devaney ◽  
◽  
Daniel M. Look
Keyword(s):  
Julia Sets ◽  
Sierpiński Curve ◽  
Sierpinski Curve

LOCALLY SIERPINSKI JULIA SETS OF WEIERSTRASS ELLIPTIC ℘ FUNCTIONS

2006 ◽  
Vol 16 (05) ◽  
pp. 1505-1520 ◽  
Author(s):  
JANE M. HAWKINS ◽  
DANIEL M. LOOK
Keyword(s):  
Elliptic Function ◽  
Fundamental Domain ◽  
Julia Set ◽  
Julia Sets ◽  
Sufficient Conditions ◽  
Elliptic Functions ◽  
Weierstrass Elliptic Function ◽  
Naturally Occurring ◽  
Sierpiński Curve ◽  
Sierpinski Curve

We define a locally Sierpinski Julia set to be a Julia set of an elliptic function which is a Sierpinski curve in each fundamental domain for the lattice. In order to construct examples, we give sufficient conditions on a lattice for which the corresponding Weierstrass elliptic ℘ function is locally connected and quadratic-like, and we use these results to prove the existence of locally Sierpinski Julia sets for certain elliptic functions. We give examples satisfying these conditions. We show this results in naturally occurring Sierpinski curves in the plane, sphere and torus as well.

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