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.

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.

scholarly journals Iteration of certain meromorphic functions with unbounded singular values

2009 ◽  
Vol 30 (3) ◽  
pp. 877-891 ◽  
Author(s):  
TARAKANTA NAYAK ◽  
M. GURU PREM PRASAD
Keyword(s):  
Real Line ◽  
Critical Parameter ◽  
Meromorphic Functions ◽  
Period Doubling ◽  
Inverse Function ◽  
Singular Values ◽  
Fatou Set ◽  
Unbounded Subset ◽  
Simply Connected ◽  
Wandering Domains

AbstractLet ℳ={f(z)=(zm/sinhm z) for z∈ℂ∣ either m or m/2 is an odd natural number}. For eachf∈ℳ, the set of singularities of the inverse function offis an unbounded subset of the real line ℝ. In this paper, the iteration of functions in one-parameter family 𝒮={fλ(z)=λf(z)∣λ∈ℝ∖{0}} is investigated for eachf∈ℳ. It is shown that, for eachf∈ℳ, there is a critical parameterλ*>0 depending onfsuch that a period-doubling bifurcation occurs in the dynamics of functionsfλin 𝒮 when the parameter |λ| passes throughλ*. The non-existence of Baker domains and wandering domains in the Fatou set offλis proved. Further, it is shown that the Fatou set offλis infinitely connected for 0<∣λ∣≤λ*whereas for ∣λ∣≥λ*, the Fatou set offλconsists of infinitely many components and each component is simply connected.

scholarly journals Wiman–Valiron Discs and the Dimension of Julia Sets

2020 ◽  
Author(s):  
James Waterman
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Julia Sets ◽  
Singular Values ◽  
Simply Connected ◽  
Set Of Points ◽  
Meromorphic Map

Abstract We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.

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 Fatou components of elliptic polynomial skew products

2017 ◽  
Vol 39 (8) ◽  
pp. 2235-2247 ◽  
Author(s):  
HAN PETERS ◽  
JASMIN RAISSY
Keyword(s):  
Julia Set ◽  
Main Tool ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Skew Products ◽  
Brjuno Condition ◽  
Key Steps ◽  
Fatou Components ◽  
Phd Thesis ◽  
Wandering Domains

We investigate the description of Fatou components for polynomial skew products in two complex variables. The non-existence of wandering domains near a super-attracting invariant fiber was shown in Lilov [Fatou theory in two dimensions. PhD Thesis, University of Michigan, 2004], and the geometrically attracting case was studied in Peters and Vivas [Polynomial skew products with wandering Fatou-disks. Math. Z.283(1–2) (2016), 349–366] and Peters and Smit [Fatou components of attracting skew products. Preprint, 2015, http://arxiv.org/abs/1508.06605]. In Astorg et al [A two-dimensional polynomial mapping with a wandering Fatou component. Ann. of Math. (2), 184 (2016), 263–313] it was proven that wandering domains can exist near a parabolic invariant fiber. In this paper we study the remaining case, namely the dynamics near an elliptic invariant fiber. We prove that the two-dimensional Fatou components near the elliptic invariant fiber correspond exactly to the Fatou components of the restriction to the fiber, under the assumption that the multiplier at the elliptic invariant fiber satisfies the Brjuno condition and that the restriction polynomial has no critical points on the Julia set. We also show the description does not hold when the Brjuno condition is dropped. Our main tool is the construction of expanding metrics on nearby fibers, and one of the key steps in this construction is given by a local description of the dynamics near a parabolic periodic cycle.

scholarly journals Dynamics of the meromorphic families $f_\lambda=\lambda \tan^pz^q$

10.53733/135 ◽  
2021 ◽  
Vol 52 ◽  
pp. 469-510
Author(s):  
Tao Chen ◽  
Linda Keen
Keyword(s):  
Meromorphic Functions ◽  
Julia Set ◽  
Singular Values ◽  
Parameter Plane ◽  
New Techniques ◽  
Asymptotic Values ◽  
The Family ◽  
Dynamical Properties ◽  
Set Of Points ◽  
Dense Set

This paper continues our investigation of the dynamics of families of transcendental meromorphic functions with finitely many singular values all of which are finite.   Here we  look at a generalization of the family of polynomials $P_a(z)=z^{d-1}(z- \frac{da}{(d-1)})$, the family $f_{\lambda}=\lambda \tan^p z^q$.  These functions have a super-attractive fixed point, and, depending on $p$, one or two asymptotic values.   Although many of the dynamical properties generalize, the existence of an essential singularity and of poles of multiplicity greater than one implies that significantly different techniques are required here.   Adding transcendental methods to standard ones, we give a description of the dynamical properties; in particular we prove the Julia set of a hyperbolic map is either connected and locally connected or a Cantor set.   We also give a description of the parameter plane of the family $f_{\lambda}$.  Again there are similarities to and differences from  the parameter plane of the family $P_a$ and again  there are new techniques.   In particular, we prove there is dense set of points on the boundaries of the hyperbolic components that are accessible along curves and we characterize these  points.

scholarly journals Classifying simply connected wandering domains

2021 ◽  
Author(s):  
Anna Miriam Benini ◽  
Vasiliki Evdoridou ◽  
Núria Fagella ◽  
Philip J. Rippon ◽  
Gwyneth M. Stallard
Keyword(s):  
Entire Functions ◽  
General Technique ◽  
Jordan Curves ◽  
Wandering Domain ◽  
Detailed Classification ◽  
Transcendental Entire Functions ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

AbstractWhile the dynamics of transcendental entire functions in periodic Fatou components and in multiply connected wandering domains are well understood, the dynamics in simply connected wandering domains have so far eluded classification. We give a detailed classification of the dynamics in such wandering domains in terms of the hyperbolic distances between iterates and also in terms of the behaviour of orbits in relation to the boundaries of the wandering domains. In establishing these classifications, we obtain new results of wider interest concerning non-autonomous forward dynamical systems of holomorphic self maps of the unit disk. We also develop a new general technique for constructing examples of bounded, simply connected wandering domains with prescribed internal dynamics, and a criterion to ensure that the resulting boundaries are Jordan curves. Using this technique, based on approximation theory, we show that all of the nine possible types of simply connected wandering domain resulting from our classifications are indeed realizable.

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 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.

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions II

2000 ◽  
Vol 20 (3) ◽  
pp. 895-910 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Statistical Mechanics ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Real Analytic ◽  
Analytic Maps

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

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