scholarly journals Homotopy Hubbard trees for post-singularly finite exponential maps

2021 ◽  
pp. 1-46
Author(s):  
DAVID PFRANG ◽  
MICHAEL ROTHGANG ◽  
DIERK SCHLEICHER
Keyword(s):  
Entire Functions ◽  
Characterization Theorem ◽  
Rational Maps ◽  
Topological Characterization ◽  
Exponential Maps ◽  
The Family ◽  
Transcendental Entire Functions ◽  
Traditional Sense ◽  
Polynomial Dynamics ◽  
Hubbard Trees

Abstract We extend the concept of a Hubbard tree, well established and useful in the theory of polynomial dynamics, to the dynamics of transcendental entire functions. We show that Hubbard trees in the strict traditional sense, as invariant compact trees embedded in $\mathbb {C}$ , do not exist even for post-singularly finite exponential maps; the difficulty lies in the existence of asymptotic values. We therefore introduce the concept of a homotopy Hubbard tree that takes care of these difficulties. Specifically for the family of exponential maps, we show that every post-singularly finite map has a homotopy Hubbard tree that is unique up to homotopy, and that post-singularly finite exponential maps can be classified in terms of homotopy Hubbard trees, using a transcendental analogue of Thurston’s topological characterization theorem of rational maps.

On the iteration of rational functions

1978 ◽  
Vol 84 (3) ◽  
pp. 497-505 ◽  
Author(s):  
V. Garber
Keyword(s):  
Rational Function ◽  
Entire Functions ◽  
Rational Functions ◽  
Transcendental Entire Function ◽  
The Family ◽  
Transcendental Entire Functions ◽  
Definition Of ◽  
Iteration Of Rational Functions ◽  
Image Position ◽  
Set Of Points

In the theory of the iteration of a rational function or transcendental entire function R(z) of the complex variable z we study the sequence of natural iterates, {Rn(z):n = 0, 1,…}, of R, whereThe domain of definition of the iterates is , the extended complex plane (if R is rational), and (if R is entire transcendental) with the topology of the chordal metric and euclidean metric respectively. Fatou(5) and Julia(9) developed a global theory of the iteration of a rational function. In (6) Fatou extended the theory of (5) to transcendental entire functions. A central role is played in the theory by the F-set, F(R), of R, R rational or entire, which is defined to be the set of points at which the family of iterates do not form a normal family in the sense of Montel.

ON EXCEPTIONAL SETS: THE SOLUTION OF A PROBLEM POSED BY K. MAHLER

2016 ◽  
Vol 94 (1) ◽  
pp. 15-19 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOSIMAR RAMIREZ
Keyword(s):  
Entire Functions ◽  
Complex Conjugation ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Transcendental Numbers ◽  
Lecture Notes ◽  
Transcendental Entire Functions

In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].

scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

scholarly journals The iteration of polynomials and transcendental entire functions

1981 ◽  
Vol 30 (4) ◽  
pp. 483-495 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Entire Functions ◽  
Unbounded Domains ◽  
Transcendental Entire Functions ◽  
Transcendental Functions

AbstractThe iterative behaviour of polynomials is contrasted with that of small transcendental functions as regards the existence of unbounded domains of normality for the sequence of iterates.

scholarly journals Dynamics on the Pre-periodic Components of the Fatou Set of Three Transcendental Entire Functions and Their Compositions

2020 ◽  
Vol 19 (1) ◽  
pp. 161-166
Author(s):  
Bishnu Hari Subedi ◽  
Ajaya Singh
Keyword(s):  
Infinite Number ◽  
Entire Functions ◽  
Periodic Component ◽  
Fatou Set ◽  
Transcendental Entire Functions ◽  
Transcendental Functions ◽  
Periodic Components

We prove that there exist three entire transcendental functions that can have an infinite number of domains which lie in the pre-periodic component of the Fatou set each of these functions and their compositions.

A new approach to the phylogeny of Euascomycetes with a cladistic outline of Arthoniales focussing on Roccellaceae

1990 ◽  
Vol 68 (11) ◽  
pp. 2458-2492 ◽  
Author(s):  
Anders Tehler
Keyword(s):  
Key Words ◽  
Type Species ◽  
Parallel Evolution ◽  
Special Focus ◽  
New Approach ◽  
Lichenicolous Fungi ◽  
Working Hypothesis ◽  
The Family ◽  
Traditional Sense ◽  
Bitunicate Ascus

A phylogenetic working hypothesis of euascomycete relationships is presented. Paraphysoidal ascolocular fungi are monophyletic and ascohymenial fungi are paraphyletic as paraphyses are considered plesiomorphic and paraphysoids apomorphic within euascomycete fungi. As a result it is not necessary to postulate parallel evolution of the bitunicate ascus, and furthermore presence of paraphyses in the prototunicate caliciaceous fungi is no longer in conflict with paraphyses in ascohymenial fungi. A cladistic outline of the order Arthoniales with special focus on the Roccellaceae including 20 taxa and 92 characters is presented. The type species of all genera considered are used as terminal taxa. It is suggested that the type species of Arthothelium should be excluded from the order Arthoniales. The Arthoniaceae are paraphyletic as Arthonia radiata and Arthothelium spectabile form a grade pair. The family Roccellaceae is monophyletic and corroborated in its traditional sense, but some rearrangements within the family are made. Opegrapha vulgata and Lecanactis abietina form a pair, but the family Opegraphaceae is paraphyletic if the pair Chiodecton sphaerale and Schismatomma pericleum are included. The originally described ascoma of the species Darbishirella gracillima, Ingaderia pulcherrima, and Reinkella fragillima are found to be lichenicolous fungi. The mycobiont ascomata of Darbishirella gracillima produce 3- not 2-septate spores. No ascomata of the mycobionts of the two latter species have as yet been found. Key words: Euascomycetes, Arthoniales, Roccellaceae, phylogeny, cladistics, lichenicolous.

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

2011 ◽  
Vol 21 (11) ◽  
pp. 3323-3339
Author(s):  
RIKA HAGIHARA ◽  
JANE HAWKINS
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Riemann Sphere ◽  
Julia Set ◽  
Critical Orbit ◽  
Rational Maps ◽  
Complex Numbers ◽  
Period 2 ◽  
The Family ◽  
Dynamical Properties

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

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