ON WANDERING AND BAKER DOMAINS OF TRANSCENDENTAL ENTIRE FUNCTIONS

2004 ◽  
Vol 14 (01) ◽  
pp. 321-327 ◽  
Author(s):  
XIAOLING WANG ◽  
CHUNG-CHUN YANG
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Fatou Component ◽  
Multiply Connected ◽  
Wandering Domain ◽  
Transcendental Entire Functions ◽  
Simply Connected

Let f denote a transcendental entire function, and I(f), I0(f), T(f) and A(f) be denoted as follows: [Formula: see text][Formula: see text] Let D denote a Fatou component of F(f). We have established the relationships between D and I(f), I0(f), T(f) or A(f), when D is a Baker domain or a multiply-connected wandering domain or a simply-connected infinitely wandering domain.

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 Fatou’s Associates

2020 ◽  
Vol 6 (3-4) ◽  
pp. 459-493
Author(s):  
Vasiliki Evdoridou ◽  
Lasse Rempe ◽  
David J. Sixsmith
Keyword(s):  
Entire Function ◽  
Blaschke Product ◽  
Entire Functions ◽  
Julia Set ◽  
Singular Values ◽  
Strong Relationship ◽  
Inner Function ◽  
Transcendental Entire Function ◽  
Fatou Component ◽  
Finite Blaschke Product

AbstractSuppose that f is a transcendental entire function, $$V \subsetneq {\mathbb {C}}$$ V ⊊ C is a simply connected domain, and U is a connected component of $$f^{-1}(V)$$ f - 1 ( V ) . Using Riemann maps, we associate the map $$f :U \rightarrow V$$ f : U → V to an inner function $$g :{\mathbb {D}}\rightarrow {\mathbb {D}}$$ g : D → D . It is straightforward to see that g is either a finite Blaschke product, or, with an appropriate normalisation, can be taken to be an infinite Blaschke product. We show that when the singular values of f in V lie away from the boundary, there is a strong relationship between singularities of g and accesses to infinity in U. In the case where U is a forward-invariant Fatou component of f, this leads to a very significant generalisation of earlier results on the number of singularities of the map g. If U is a forward-invariant Fatou component of f there are currently very few examples where the relationship between the pair (f, U) and the function g has been calculated. We study this relationship for several well-known families of transcendental entire functions. It is also natural to ask which finite Blaschke products can arise in this manner, and we show the following: for every finite Blaschke product g whose Julia set coincides with the unit circle, there exists a transcendental entire function f with an invariant Fatou component such that g is associated with f in the above sense. Furthermore, there exists a single transcendental entire function f with the property that any finite Blaschke product can be arbitrarily closely approximated by an inner function associated with the restriction of f to a wandering domain.

scholarly journals Some entire functions with multiply-connected wandering domains

1985 ◽  
Vol 5 (2) ◽  
pp. 163-169 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Entire Function ◽  
Entire Functions ◽  
Julia Set ◽  
Multiply Connected ◽  
Wandering Domain ◽  
Wandering Domains

AbstractA component U of the complement of the Julia set of an entire function ƒ is a wandering domain if the sets ƒn(U) are mutually disjoint, where n ∈ℕ and ƒn is the n-th iterate of ƒ. Examples are given of entire ƒ of order , which have multiply-connected wandering domains. An example is given where the connectivity is infinite.

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 Hollow quasi-Fatou components of quasiregular maps

2016 ◽  
Vol 162 (3) ◽  
pp. 561-574
Author(s):  
DANIEL A. NICKS ◽  
DAVID J. SIXSMITH
Keyword(s):  
Entire Function ◽  
Julia Set ◽  
The Other ◽  
Transcendental Entire Function ◽  
Connected Component ◽  
Fatou Component ◽  
Multiply Connected ◽  
Complementary Component ◽  
Fatou Components ◽  
Isolated Point

AbstractWe define a quasi-Fatou component of a quasiregular map as a connected component of the complement of the Julia set. A domain in ℝd is called hollow if it has a bounded complementary component. We show that for each d ⩾ 2 there exists a quasiregular map of transcendental type f: ℝd → ℝd with a quasi-Fatou component which is hollow.Suppose that U is a hollow quasi-Fatou component of a quasiregular map of transcendental type. We show that if U is bounded, then U has many properties in common with a multiply connected Fatou component of a transcendental entire function. On the other hand, we show that if U is not bounded, then it is completely invariant and has no unbounded boundary components. We show that this situation occurs if J(f) has an isolated point, or if J(f) is not equal to the boundary of the fast escaping set. Finally, we deduce that if J(f) has a bounded component, then all components of J(f) are bounded.

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.

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.

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.

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