Iterates of meromorphic functions: I

1991 ◽  
Vol 11 (2) ◽  
pp. 241-248 ◽  
Author(s):  
I. N. Baker ◽  
J. Kotus ◽  
Lü Yinian
Keyword(s):  
Meromorphic Functions ◽  
Entire Functions ◽  
Julia Set ◽  
Exceptional Case ◽  
Straight Line ◽  
Transcendental Entire Functions ◽  
Periodic Cycles

AbstractFor functions meromorphic in the plane, apart from an exceptional case, the Julia set J is the closure of the set of all preimages of poles. The repelling periodic cycles are dense in J. In contrast with the case of transcendental entire functions, J may be a subset of a straight line and general classes of functions for which this is the case can be determined. J may also lie on a quasicircle through infinity which is not a straight line.

scholarly journals Hausdorff measure of escaping and Julia sets for bounded-type functions of finite order

2011 ◽  
Vol 33 (1) ◽  
pp. 284-302 ◽  
Author(s):  
JÖRN PETER
Keyword(s):  
Finite Order ◽  
Hausdorff Measure ◽  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Gauge Function ◽  
Exponential Map ◽  
Hausdorff Measures ◽  
Transcendental Entire Functions ◽  
Bounded Type Functions

AbstractWe show that the escaping sets and the Julia sets of bounded-type transcendental entire functions of order ρ become ‘smaller’ as ρ→∞. More precisely, their Hausdorff measures are infinite with respect to the gauge function hγ(t)=t2g(1/t)γ, where g is the inverse of a linearizer of some exponential map and γ≥(log ρ(f)+K1)/c, but for ρ large enough, there exists a function fρ of bounded type with order ρ such that the Hausdorff measures of the escaping set and the Julia set of fρ with respect to hγ′ are zero whenever γ′ ≤(log ρ−K2)/c.

scholarly journals Maximally and non-maximally fast escaping points of transcendental entire functions

2015 ◽  
Vol 158 (2) ◽  
pp. 365-383 ◽  
Author(s):  
D. J. SIXSMITH
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Julia Set ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions ◽  
Dynamical Properties

AbstractWe partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest.It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire function for which the maximally and non-maximally fast escaping sets each have uncountably many singleton components.

scholarly journals Orbifold expansion and entire functions with bounded Fatou components

2021 ◽  
pp. 1-40
Author(s):  
LETICIA PARDO-SIMÓN
Keyword(s):  
General Class ◽  
Entire Functions ◽  
Julia Set ◽  
Holomorphic Dynamics ◽  
Fatou Set ◽  
Local Connectivity ◽  
Hyperbolic Functions ◽  
Hyperbolic Orbifold ◽  
Transcendental Entire Functions ◽  
Fatou Components

Abstract Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

scholarly journals Dynamics of slowly growing entire functions

2001 ◽  
Vol 63 (3) ◽  
pp. 367-377 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Entire Function ◽  
Normal Family ◽  
Entire Functions ◽  
Julia Set ◽  
Maximum Modulus ◽  
Stable Set ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions ◽  
Transcendental Functions ◽  
Simply Connected

Dedicated to George Szekeres on his 90th birthdayFor a transcendental entire function f let M(r) denote the maximum modulus of f(z) for |z| = r. Then A(r) = log M(r)/logr tends to infinity with r. Many properties of transcendental entire functions with sufficiently small A(r) resemble those of polynomials. However the dynamical properties of iterates of such functions may be very different. For instance in the stable set F(f) where the iterates of f form a normal family the components are preperiodic under f in the case of a polynomial; but there are transcendental functions with arbitrarily small A(r) such that F(f) has nonpreperiodic components, so called wandering components, which are bounded rings in which the iterates tend to infinity. One might ask if all small functions are like this.A striking recent result of Bergweiler and Eremenko shows that there are arbitrarily small transcendental entire functions with empty stable set—a thing impossible for polynomials. By extending the technique of Bergweiler and Eremenko, an arbitrarily small transcendental entire function is constructed such that F is nonempty, every component G of F is bounded, simply-connected and the iterates tend to zero in G. Zero belongs to an invariant component of F, so there are no wandering components. The Julia set which is the complement of F is connected and contains a dense subset of “buried’ points which belong to the boundary of no component of F. This bevaviour is impossible for a polynomial.

The Hausdorff dimension of Julia sets of entire functions III

1997 ◽  
Vol 122 (2) ◽  
pp. 223-244 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Entire Function ◽  
Hausdorff Dimension ◽  
Entire Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions

It is known that, for a transcendental entire function f, the Hausdorff dimension of the Julia set of f satisfies 1[les ]dim J(f)[les ]2. In this paper we introduce a family of transcendental entire functions fp, K for which the set {dim J(fp, K)[ratio ]0<p, K<∞} has infemum 1 and supremum 2.

scholarly journals Spiders’ webs and locally connected Julia sets of transcendental entire functions

2012 ◽  
Vol 33 (4) ◽  
pp. 1146-1161 ◽  
Author(s):  
J. W. OSBORNE
Keyword(s):  
Entire Function ◽  
Opposite Direction ◽  
Entire Functions ◽  
Dense Subset ◽  
Julia Set ◽  
Julia Sets ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions

AbstractWe show that if the Julia set of a transcendental entire function is locally connected, then it takes the form of a spider’s web in the sense defined by Rippon and Stallard. In the opposite direction, we prove that a spider’s web Julia set is always locally connected at a dense subset of buried points. We also show that the set of buried points (the residual Julia set) can be a spider’s web.

scholarly journals Criniferous entire maps with absorbing Cantor bouquets

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Leticia Pardo-Simón
Keyword(s):  
Entire Functions ◽  
Julia Set ◽  
New Class ◽  
Transcendental Entire Functions

<p style='text-indent:20px;'>It is known that, for many transcendental entire functions in the Eremenko-Lyubich class <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>, every escaping point can eventually be connected to infinity by a curve of escaping points. When this is the case, we say that the functions are <i>criniferous</i>. In this paper, we extend this result to a new class of maps in <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula>. Furthermore, we show that if a map belongs to this class, then its Julia set contains a <i>Cantor bouquet</i>; in other words, it is a subset of <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{C} $\end{document}</tex-math></inline-formula> ambiently homeomorphic to a straight brush.</p>

scholarly journals Permutable functions concerning differential equations

2007 ◽  
Vol 83 (3) ◽  
pp. 369-384 ◽  
Author(s):  
X. Hua ◽  
R. Vaillancourt ◽  
X. L. Wang
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Entire Functions ◽  
Julia Set ◽  
Rational Functions ◽  
Exponential Functions ◽  
Polynomial Coefficients ◽  
Transcendental Entire Functions ◽  
Linear Differential ◽  
Open Question

AbstractLet f and g be two permutable transcendental entire functions. Assume that f is a solution of a linear differential equation with polynomial coefficients. We prove that, under some restrictions on the coefficients and the growth of f and g, there exist two non-constant rational functions R1 and R2 such that R1 (f) = R(g). As a corollary, we show that f and g have the same Julia set: J(f) = J(g). As an application, we study a function f which is a combination of exponential functions with polynomial coefficients. This research addresses an open question due to Baker.

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)].

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