The Dynamics of Planar Quadratic Maps with Nonempty Bounded Critical Set

1998 ◽  
Vol 08 (01) ◽  
pp. 95-105 ◽  
Author(s):  
Chia-Hsing Nien
Keyword(s):  
Julia Sets ◽  
Periodic Points ◽  
Critical Set ◽  
Quadratic Map ◽  
Quadratic Maps

If the critical set of a planar quadratic map is bounded and nonempty, then the range is doubly covered except the image of the critical set and the region bounded by it (Fig. 1). For a map of this type, we showed that infinity is a sink and the set of bounded orbits is nonempty hence its boundary may be regarded as a generalization of Julia sets of complex quadratic maps. In this paper, we also provided evidences to conjecture that maps of this type have infinitely many periodic points.

scholarly journals The -limit sets of quadratic Julia sets

2013 ◽  
Vol 35 (2) ◽  
pp. 337-358 ◽  
Author(s):  
ANDREW D. BARWELL ◽  
BRIAN E. RAINES
Keyword(s):  
Julia Sets ◽  
Symbolic Representation ◽  
Invariant Set ◽  
Limit Set ◽  
Limit Sets ◽  
Quadratic Maps

AbstractIn this paper we characterize $\omega $-limit sets of dendritic Julia sets for quadratic maps. We use Baldwin’s symbolic representation of these spaces as a non-Hausdorff itinerary space and prove that quadratic maps with dendritic Julia sets have shadowing, and also that for all such maps, a closed invariant set is an $\omega $-limit set of a point if, and only if, it is internally chain transitive.

scholarly journals Quasisymmetric conjugacy between quadratic dynamics and iterated function systems

2009 ◽  
Vol 30 (6) ◽  
pp. 1665-1684 ◽  
Author(s):  
KEMAL ILGAR EROĞLU ◽  
STEFFEN ROHDE ◽  
BORIS SOLOMYAK
Keyword(s):  
Julia Set ◽  
Iterated Function Systems ◽  
Critical Set ◽  
Simple Modification ◽  
Quadratic Map ◽  
Finite Case ◽  
Contraction Ratio ◽  
Iterated Function ◽  
The Family ◽  
Continuous Surjection

AbstractWe consider linear iterated function systems (IFS) with a constant contraction ratio in the plane for which the ‘overlap set’ 𝒪 is finite, and which are ‘invertible’ on the attractor A, in the sense that there is a continuous surjection q:A→A whose inverse branches are the contractions of the IFS. The overlap set is the critical set in the sense that q is not a local homeomorphism precisely at 𝒪. We suppose also that there is a rational function p with the Julia set J such that (A,q) and (J,p) are conjugate. We prove that if A has bounded turning and p has no parabolic cycles, then the conjugacy is quasisymmetric. This result is applied to some specific examples including an uncountable family. Our main focus is on the family of IFS {λz,λz+1} where λ is a complex parameter in the unit disk, such that its attractor Aλ is a dendrite, which happens whenever 𝒪 is a singleton. C. Bandt observed that a simple modification of such an IFS (without changing the attractor) is invertible and gives rise to a quadratic-like map qλ on Aλ. If the IFS is post-critically finite, then a result of A. Kameyama shows that there is a quadratic map pc (z)=z2 +c, with the Julia set Jc such that (Aλ,qλ) and (Jc,pc) are conjugate. We prove that this conjugacy is quasisymmetric and obtain partial results in the general (not post-critically finite) case.

Wreath products along the period-doubling route to chaos

1999 ◽  
Vol 19 (6) ◽  
pp. 1617-1636 ◽  
Author(s):  
J. D. H. SMITH
Keyword(s):  
Wreath Product ◽  
Period Doubling ◽  
Periodic Points ◽  
Wreath Products ◽  
Explicit Description ◽  
Quadratic Maps ◽  
Combinatorial Description ◽  
Product Construction ◽  
Renormalization Operator ◽  
Route To Chaos

The wreath-product construction is used to give a complete combinatorial description of the dynamics of period-doubling quadratic maps leading to the Feigenbaum map. An explicit description of the action on periodic points uses the Thue–Morse sequence. In particular, a wreath-product construction of this sequence is given. The combinatorial renormalization operator on the period-doubling family of maps is invertible.

Geometry and combinatorics of Julia sets of real quadratic maps

1984 ◽  
Vol 37 (1-2) ◽  
pp. 51-92 ◽  
Author(s):  
M. F. Barnsley ◽  
J. S. Geronimo ◽  
A. N. Harrington
Keyword(s):  
Julia Sets ◽  
Quadratic Maps ◽  
Real Quadratic

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.

scholarly journals Julia sets of Zorich maps

2021 ◽  
pp. 1-37
Author(s):  
ATHANASIOS TSANTARIS
Keyword(s):  
Complex Plane ◽  
Pairwise Disjoint ◽  
Exponential Family ◽  
Julia Set ◽  
Julia Sets ◽  
Three Dimensional ◽  
Periodic Points ◽  
Exponential Map ◽  
Simple Curves ◽  
Entire Complex

Abstract The Julia set of the exponential family $E_{\kappa }:z\mapsto \kappa e^z$ , $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<\kappa \leq 1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three-dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the whole of $\mathbb {R}^3$ , generalizing Misiurewicz’s result. Moreover, we show that the periodic points of the Zorich map are dense in $\mathbb {R}^3$ and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.

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