scholarly journals ON THE UNIFORM PERFECTNESS OF THE BOUNDARY OF MULTIPLY CONNECTED WANDERING DOMAINS

2011 ◽  
Vol 91 (3) ◽  
pp. 289-311 ◽  
Author(s):  
WALTER BERGWEILER ◽  
JIAN-HUA ZHENG
Keyword(s):  
Entire Function ◽  
General Criterion ◽  
Multiply Connected ◽  
Uniformly Perfect ◽  
Wandering Domain ◽  
Uniform Perfectness ◽  
Wandering Domains

AbstractWe investigate when the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.

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.

On uniformly perfect boundary of stable domains in iteration of meromorphic functions II

2002 ◽  
Vol 132 (3) ◽  
pp. 531-544 ◽  
Author(s):  
ZHENG JIAN-HUA
Keyword(s):  
Entire Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Fatou Set ◽  
Deficient Value ◽  
Uniformly Perfect ◽  
Simply Connected ◽  
Stable Domains ◽  
Uniform Perfectness

We investigate uniform perfectness of the Julia set of a transcendental meromorphic function with finitely many poles and prove that the Julia set of such a meromorphic function is not uniformly perfect if it has only bounded components. The Julia set of an entire function is uniformly perfect if and only if the Julia set including infinity is connected and every component of the Fatou set is simply connected. Furthermore if an entire function has a finite deficient value in the sense of Nevanlinna, then it has no multiply connected stable domains. Finally, we give some examples of meromorphic functions with uniformly perfect Julia sets.

scholarly journals An entire function which has wandering domains

1976 ◽  
Vol 22 (2) ◽  
pp. 173-176 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Entire Function ◽  
Complex Variable ◽  
Normal Family ◽  
Proper Subset ◽  
Transcendental Entire Function ◽  
Wandering Domain ◽  
Wandering Domains

AbstractLet f(z) denote a rational or entire function of the complex variable z and fn(z), n = 1,2, …, the n−th iterate of f. Provided f is not rational of order 0 or 1, the set of those points where {fn(z)} forms a normal family is a proper subset of the plane and is invariant under the map z → f(z). A component G of is a wandering domain of f if fn(G)∩fn(G) = Ø for all k ≧ 1, n ≧ 1, k ≠ n. The paper contains the construction of a transcendental entire function which has wandering domains.

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 Escaping Fatou components of transcendental self-maps of the punctured plane

2019 ◽  
pp. 1-25 ◽  
Author(s):  
DAVID MARTÍ-PETE
Keyword(s):  
Approximation Theory ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Sufficient Condition ◽  
Wandering Domain ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

Abstract We study the iteration of transcendental self-maps of $\,\mathbb{C}^*\!:=\mathbb{C}\setminus \{0\}$ , that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\,\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

Diffeomorphisms of the k-torus with wandering domains

1995 ◽  
Vol 15 (6) ◽  
pp. 1189-1205 ◽  
Author(s):  
Patrick D. McSwiggen
Keyword(s):  
Small Perturbation ◽  
Dense Orbit ◽  
Wandering Domain ◽  
Wandering Domains

AbstractIt is shown that diffeomorphisms analogous to a classical example on the circle due to Denjoy can be constructed on the general k-torus. Such a diffeomorphism has the property that it is semiconjugate to an ergodic translation but has a wandering domain with dense orbit. The construction on the k-torus can be made Cr, and by a Cr small perturbation of a translation, for any r < k + 1.

Hyperbolic Metric and Multiply Connected Wandering Domains of Meromorphic Functions

2017 ◽  
Vol 60 (3) ◽  
pp. 787-810
Author(s):  
Jian-Hua Zheng
Keyword(s):  
Analytic Function ◽  
Meromorphic Function ◽  
Meromorphic Functions ◽  
Hyperbolic Metric ◽  
Multiply Connected ◽  
The Hyperbolic Metric ◽  
Large Round ◽  
Hyperbolic Domain ◽  
Wandering Domains

AbstractIn this paper, in terms of the hyperbolic metric, we give a condition under which the image of a hyperbolic domain of an analytic function contains a round annulus centred at the origin. From this, we establish results on the multiply connected wandering domains of a meromorphic function that contain large round annuli centred at the origin. We thereby successfully extend the results of transcendental meromorphic functions with finitely many poles to those with infinitely many poles.

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 Fatou components and singularities of meromorphic functions

2019 ◽  
Vol 150 (2) ◽  
pp. 633-654 ◽  
Author(s):  
Krzysztof Barański ◽  
Núria Fagella ◽  
Xavier Jarque ◽  
Bogusława Karpińska
Keyword(s):  
Relative Position ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Singular Values ◽  
Wandering Domain ◽  
Meromorphic Maps ◽  
Fatou Components ◽  
Meromorphic Map ◽  
Wandering Domains ◽  
Uniformly Bounded

AbstractWe prove several results concerning the relative position of points in the postsingular set P(f) of a meromorphic map f and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates Un of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values pn such that ${\rm dist} (p_n, U_n)\to 0$ as $n\to \infty $. We also prove that if $U_n \cap P(f)=\emptyset $ and the postsingular set of f lies at a positive distance from the Julia set (in ℂ), then the sequence of iterates of any wandering domain must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

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