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.

scholarly journals ON SOLUTIONS TO NONHOMOGENEOUS ALGEBRAIC DIFFERENTIAL EQUATIONS AND THEIR APPLICATION

2014 ◽  
Vol 97 (3) ◽  
pp. 391-403 ◽  
Author(s):  
LIANG-WEN LIAO ◽  
ZHUAN YE
Keyword(s):  
Differential Equation ◽  
Entire Function ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Meromorphic Solution ◽  
Differential Polynomial ◽  
Transcendental Entire Function ◽  
Algebraic Differential Equation ◽  
Algebraic Differential Equations ◽  
Image Position

AbstractWe consider solutions to the algebraic differential equation $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$, where $Q_d(z,f)$ is a differential polynomial in $f$ of degree $d$ with rational function coefficients, $u$ is a nonzero rational function and $v$ is a nonconstant polynomial. In this paper, we prove that if $n\ge d+1$ and if it admits a meromorphic solution $f$ with finitely many poles, then $$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$ With this in hand, we also prove that if $f$ is a transcendental entire function, then $f'p_k(f)+q_m(f)$ assumes every complex number $\alpha $, with one possible exception, infinitely many times, where $p_k(f), q_m(f)$ are polynomials in $f$ with degrees $k$ and $m$ with $k\ge m+1$. This result generalizes a theorem originating from Hayman [‘Picard values of meromorphic functions and their derivatives’, Ann. of Math. (2)70(2) (1959), 9–42].

On the Taylor coefficients of rational functions

1956 ◽  
Vol 52 (1) ◽  
pp. 39-48 ◽  
Author(s):  
K. Mahler ◽  
J. W. S. Cassels
Keyword(s):  
Power Series ◽  
Rational Function ◽  
Rational Functions ◽  
Partial Solution ◽  
Convergent Power Series ◽  
Taylor Coefficients ◽  
Image Position

Let F(z) be a rational function of z which is regular at z = 0 and so possesses a convergent power seriesThe problem arises of characterizing those rational functions F(z) that have infinitely many vanishing Taylor coefficientsfh. After earlier and more special results by Siegel(2) and Ward(4) I applied in 1934(1) a p-adic method due to Skolem(3) to the problem and obtained the following partial solution.

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 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.

scholarly journals Dynamical sets whose union with infinity is connected

2018 ◽  
Vol 40 (3) ◽  
pp. 789-798 ◽  
Author(s):  
DAVID J. SIXSMITH
Keyword(s):  
Entire Function ◽  
Large Class ◽  
Topological Structure ◽  
Entire Functions ◽  
Sufficient Conditions ◽  
Related Question ◽  
Transcendental Entire Function ◽  
Transcendental Entire Functions ◽  
Class Of Functions

Suppose that $f$ is a transcendental entire function. In 2011, Rippon and Stallard showed that the union of the escaping set with infinity is always connected. In this paper we consider the related question of whether the union with infinity of the bounded orbit set, or the bungee set, can also be connected. We give sufficient conditions for these sets to be connected and an example of a transcendental entire function for which all three sets are simultaneously connected. This function lies, in fact, in the Speiser class.It is known that for many transcendental entire functions the escaping set has a topological structure known as a spider’s web. We use our results to give a large class of functions in the Eremenko–Lyubich class for which the escaping set is not a spider’s web. Finally, we give a novel topological criterion for certain sets to be a spider’s web.

ON THE PERIODICITY OF TRANSCENDENTAL ENTIRE FUNCTIONS

2020 ◽  
Vol 101 (3) ◽  
pp. 453-465 ◽  
Author(s):  
XINLING LIU ◽  
RISTO KORHONEN
Keyword(s):  
Entire Function ◽  
Positive Integer ◽  
Periodic Function ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Difference Polynomials ◽  
Transcendental Entire Functions ◽  
Hyper Order

According to a conjecture by Yang, if $f(z)f^{(k)}(z)$ is a periodic function, where $f(z)$ is a transcendental entire function and $k$ is a positive integer, then $f(z)$ is also a periodic function. We propose related questions, which can be viewed as difference or differential-difference versions of Yang’s conjecture. We consider the periodicity of a transcendental entire function $f(z)$ when differential, difference or differential-difference polynomials in $f(z)$ are periodic. For instance, we show that if $f(z)^{n}f(z+\unicode[STIX]{x1D702})$ is a periodic function with period $c$, then $f(z)$ is also a periodic function with period $(n+1)c$, where $f(z)$ is a transcendental entire function of hyper-order $\unicode[STIX]{x1D70C}_{2}(f)<1$ and $n\geq 2$ is an integer.

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.

On the Hausdorff measure of Brownian paths in the plane

1961 ◽  
Vol 57 (2) ◽  
pp. 209-222 ◽  
Author(s):  
P. Erdős ◽  
S. J. Taylor
Keyword(s):  
Mathematical Model ◽  
Probability Measure ◽  
Hausdorff Measure ◽  
Present Note ◽  
Continuous Curve ◽  
Borel Field ◽  
Point Set ◽  
Definition Of ◽  
Image Position ◽  
Set Of Points

Ω will denote the space of all plane paths ω, so that ω is a short way of denoting the curve . we assume that there is a probability measure μ defined on a Borel field of (measurable) subsets of Ω, so that the system (Ω, , μ) forms a mathematical model for Brownian paths in the plane. [For details of the definition of μ, see for example (9).] let L(a, b; μ) be the plane set of points z(t, ω) for a≤t≤b. Then with probability 1, L(a, b; μ) is a continuous curve in the plane. The object of the present note is to consider the measure of this point set L(a, b; ω).

Pairs of non-homogeneous linear differential polynomials

2006 ◽  
Vol 136 (4) ◽  
pp. 785-794 ◽  
Author(s):  
J. K. Langley
Keyword(s):  
Differential Equations ◽  
Rational Function ◽  
Rational Functions ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Linear Differential Polynomials ◽  
Linear Differential ◽  
Image Position ◽  
Homogeneous Linear

Let f be transcendental and meromorphic in the plane and let the non-homogeneous linear differential polynomials F and G be defined by where k,n ∈ N and a, b and the aj, bj are rational functions. Under the assumption that F and G have few zeros, it is shown that either F and G reduce to homogeneous linear differential polynomials in f + c, where c is a rational function that may be computed explicitly, or f has a representation as a rational function in solutions of certain associated linear differential equations, which again may be determined explicitly from the aj, bj and a and b.

scholarly journals A Review on the Structure and Properties of the Escaping Set of Transcendental Entire Functions

2016 ◽  
Vol 34 (1-2) ◽  
pp. 65-78
Author(s):  
Bishnu Hari Subedi ◽  
Ajaya Singh
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Structure And Properties ◽  
Closed Sets ◽  
Transcendental Entire Functions

For a transcendental entire function f, we study the structure and properties of the escaping set I(f) which consists of points whose iterates under f escape to infinity. We concentrate on Eremenko’s conjecture and we review some attempts of its proofs. A significant amount of progress in Eremenko’s conjecture has been made possible via fast escaping set A(f) which consists points that escape to infinity as fast as possible. This set can be written as union of closed sets, called levels of A(f). We review classes of functions for which A(f) and each of its levels has the structure of infinite spider’s web. In general, we study classes of entire functions for which the escaping set I(f) is a spider’s web. Spider’s web is a recently investigated structure of I(f) that gives new results in the direction of Eremenko’s conjecture.

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