Wandering domains in the transcendental dynamics

2001 ◽  
Vol 46 (1) ◽  
pp. 34-35
Author(s):  
Yuefei Wang
Keyword(s):  
Wandering Domains ◽  
Transcendental Dynamics

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.

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 Measure and Capacity of Wandering Domains in Gevrey Near-integrable Exact Symplectic Systems

2019 ◽  
Vol 257 (1235) ◽  
pp. 0-0 ◽  
Author(s):  
Laurent Lazzarini ◽  
Jean-Pierre Marco ◽  
David Sauzin
Keyword(s):  
Wandering Domains
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