Misiurewicz Points for Complex Exponentials

1997 ◽  
Vol 07 (07) ◽  
pp. 1599-1615 ◽  
Author(s):  
Robert L. Devaney ◽  
Xavier Jarque
Keyword(s):  
Symbolic Dynamics ◽  
Topological Structure ◽  
Julia Set ◽  
Chaotic Regime ◽  
Singular Value ◽  
Entire Plane ◽  
Exponential Maps ◽  
Complex Exponential

In this paper we examine the structure of the chaotic regime or Julia set of certain complex exponential maps Eλ(z) = λez. In the case where λ is a Misiurewicz point (i.e. the singular value 0 is eventually periodic), it is known that the Julia set for the map is the entire plane. In this case the Julia set also possesses certain curves or "hairs" that are permuted by the map. We examine the dynamics on these hairs in detail. We describe a certain extended symbolic dynamics by which the topological structure of the hairs may be determined completely.

Knaster-like continua and complex dynamics

1993 ◽  
Vol 13 (4) ◽  
pp. 627-634 ◽  
Author(s):  
Robert L. Devaney
Keyword(s):  
Complex Dynamics ◽  
Julia Set ◽  
Dynamical Behavior ◽  
Invariant Sets ◽  
Entire Plane ◽  
Limit Sets

AbstractIn this paper we discuss the topology and dynamics ofEλ(z) = λezwhen λ is real and λ > 1/e. It is known that the Julia set ofEλis the entire plane in this case. Our goal is to show that there are certain natural invariant subsets forEλwhich are topologically Knaster-like continua. Moreover, the dynamical behavior on these invariant sets is quite tame. We show that the only trivial kinds of α- and ω-limit sets are possible.

scholarly journals Dynamics of exp (z)

1984 ◽  
Vol 4 (1) ◽  
pp. 35-52 ◽  
Author(s):  
Robert L. Devaney ◽  
Michal Krych
Keyword(s):  
Complex Plane ◽  
Symbolic Dynamics ◽  
Final Section ◽  
Julia Set ◽  
Dynamical Behaviour ◽  
Transcendental Function ◽  
Entire Transcendental Function ◽  
Orbit Structure ◽  
The Family ◽  
Entire Complex

AbstractWe describe the dynamical behaviour of the entire transcendental function exp(z). We use symbolic dynamics to describe the complicated orbit structure of this map whose Julia Set is the entire complex plane. Bifurcations occurring in the family c exp(z) are discussed in the final section.

Symbolic Dynamics for IFS Attractors

Fractals ◽  
1997 ◽  
Vol 05 (02) ◽  
pp. 237-246 ◽  
Author(s):  
Sonya Bahar
Keyword(s):  
Symbolic Dynamics ◽  
Iterated Function System ◽  
Chaotic Regime ◽  
Function System ◽  
Control Parameters ◽  
Closed Orbits ◽  
Iterated Function ◽  
Parameter Values

It has recently been shown that a modified iterated function system (IFS) is capable of generating closed orbits which undergo bifurcation and transition to a chaotic regime as control parameters are varied.1,2 Here we show that driving such an IFS by a partition of itself creates maps which can be characterized by a symbolic dynamics. Forbidden words are determined for this dynamics under various parameter values, and the implications of this mapping are discussed.

Tying hairs for structurally stable exponentials

2000 ◽  
Vol 20 (6) ◽  
pp. 1603-1617 ◽  
Author(s):  
RANJIT BHATTACHARJEE ◽  
ROBERT L. DEVANEY
Keyword(s):  
Fixed Point ◽  
Julia Set ◽  
Basins Of Attraction ◽  
Crucial Importance ◽  
Exponential Functions ◽  
Complex Exponential ◽  
Structurally Stable

Our goal in this paper is to describe the structure of the Julia set of complex exponential functions that possess an attracting cycle. When the cycle is a fixed point, it is known that the Julia set is a ‘Cantor bouquet’, a union of uncountably many distinct curves or ‘hairs’. When the period of the cycle is greater than one, infinitely many of the hairs in the bouquet become pinched or attached together. In this paper, we develop an algorithm to determine which of these hairs are attached. Of crucial importance in this construction is the kneading invariant, a sequence that is derived from the topology of the basins of attraction of the attracting cycle.

Hausdorff dimension of hairs and ends for entire maps of finite order

2008 ◽  
Vol 145 (3) ◽  
pp. 719-737 ◽  
Author(s):  
KRZYSZTOF BARAŃSKI
Keyword(s):  
Fixed Point ◽  
Hausdorff Dimension ◽  
Finite Order ◽  
Compact Subset ◽  
Julia Set ◽  
Unique Point ◽  
Bounded Set ◽  
Exponential Maps ◽  
Attracting Fixed Point

AbstractWe study transcendental entire mapsfof finite order, such that all the singularities off−1are contained in a compact subset of the immediate basinBof an attracting fixed point off. Then the Julia set offconsists of disjoint curves tending to infinity (hairs), attached to the unique point accessible fromB(endpoint of the hair). We prove that the Hausdorff dimension of the set of endpoints of the hairs is equal to 2, while the union of the hairs without endpoints has Hausdorff dimension 1, which generalizes the result for exponential maps. Moreover, we show that for every transcendental entire map of finite order from class(i.e. with bounded set of singularities) the Hausdorff dimension of the Julia set is equal to 2.

scholarly journals An explosion point for the set of endpoints of the Julia set of λ exp (z)

1990 ◽  
Vol 10 (1) ◽  
pp. 177-183 ◽  
Author(s):  
John C. Mayer
Keyword(s):  
Exponential Function ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Julia Set ◽  
Periodic Points ◽  
Real Parameter ◽  
Topological Dimension ◽  
Complex Exponential ◽  
Totally Disconnected ◽  
Topological Sense

AbstractThe Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to ∞. Since Aλ contains all the repelling periodic points of Eλ, it follows that Jλ = Cl (Aλ). We show that Aλ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, Aλ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.

Characterization of Dynamics of a Class of Polynomial Automorphisms in ℂn

2019 ◽  
Vol 29 (01) ◽  
pp. 1950007
Author(s):  
Xu Zhang
Keyword(s):  
Symbolic Dynamics ◽  
Julia Set ◽  
Complex Polynomial ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Polynomial Mappings ◽  
Bounded Set ◽  
Higher Dimensional ◽  
Projective Limits

A kind of higher-dimensional complex polynomial mappings [Formula: see text] is considered: [Formula: see text] where [Formula: see text], [Formula: see text] are polynomials with degrees higher than one, and [Formula: see text] are nonzero complex numbers, [Formula: see text]. Assume that each [Formula: see text] is hyperbolic on its Julia set and [Formula: see text] is sufficiently small, [Formula: see text], then there exists a bounded set on which the dynamics on the forward and backwards Julia sets are described by using the inductive and the projective limits, respectively. These results are a natural higher-dimensional generalization of the work of Hubbard and Oberste-Vorth on two-dimensional complex Hénon mappings. The combination of the symbolic dynamics and the crossed mapping is also applied to study the complicated dynamics of a class of polynomial mappings in [Formula: see text].

JULIA SET OF THE NEWTON METHOD FOR SOLVING SOME COMPLEX EXPONENTIAL EQUATION

2009 ◽  
Vol 09 (02) ◽  
pp. 153-169 ◽  
Author(s):  
XINGYUAN WANG ◽  
WENJING SONG ◽  
LIXIAN ZOU
Keyword(s):  
Newton Method ◽  
Exponential Function ◽  
Julia Set ◽  
Julia Sets ◽  
Exponential Equation ◽  
Distribution Structure ◽  
Complex Exponential

We extend Kim's complex exponential function and come up with a theory about Julia sets of Newton method for general exponential equation. We analyze the behavior of the roots of some complex exponential equation, and prove the Julia Set's symmetry, boundedness and embedding topology distribution structure of attraction regions in theory.

ez: DYNAMICS AND BIFURCATIONS

1991 ◽  
Vol 01 (02) ◽  
pp. 287-308 ◽  
Author(s):  
ROBERT L. DEVANEY
Keyword(s):  
Parameter Space ◽  
Symbolic Dynamics ◽  
Exponential Family ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Structural Instability ◽  
Entire Complex ◽  
Parameter Values ◽  
The Relationship ◽  
Complex Exponential

In this paper we describe some of the dynamical behavior of the complex exponential λ exp z. For various values of λ, this family exhibits chaotic behavior on the entire complex plane. For other values, the dynamics are relatively tame. We show how to analyze this behavior via symbolic dynamics and investigate the structural instability at various parameter values. Finally, we describe the relationship between the parameter space for the exponential family and the related families of polynomials given by [Formula: see text].

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