Periodic Fatou Components of Meromorphic Functions

In iteration theory of rational functions, it is well knownthat any Fatou component is mapped onto another in an n-to-1manner, and that periodic components are simply, doubly or infinitelyconnected. For meromorphic functions, the situation is much more complicated.Using Ahlfors' theory of covering surfaces, we prove that Fatoucomponents are mapped ‘nearly’ onto others, andthat periodic components are again simply, doubly or infinitelyconnected. Instead of considering meromorphic functions withonly one essential singularity, we allow countable sets of singularitiesand partly even sets of logarithmic capacity zero. It remains open whether doubly connected periodic componentsof meromorphic functions with only one singularity are necessarilyHerman rings (as holds for rational functions). However, thereis a function with two singularities and a doubly connectedperiodic component which is not an Herman ring. 1991 MathematicsSubject Classification 30D05, 30D30.     

On the Simple Connectivity of Fatou Components

We shall show that for certain holomorphic maps, all Fatou components are simply connected. We also discuss the relation between wandering domains and singularities for certain meromorphic maps. Received May 8, 2002, Accepted May 24, 2002     

有理映照Fatou分支的连通数

Baker曾用拟共形手术的方法证明了具有任意连通数的Fatou分支的存在性,Shishikua曾建议对此给出明确的有理映照的例子,本文在Beardon工作的基础上,给出了比较完善的结果。     

超越整函数Fatou分支的某些性质

讨论了有限个超越整函数fi(i=1,2,…,m}迭代生成的函数g=f1 o f2 o…o fm 的Fatou分支的性质,给出了g(z)的Fatou分支有界的一个充分条件.并证明了在该条件下,对g的任意游荡域U,都有{gn}的一个子列在U上趋于∞.     

缺项幂级数表示的超越整函数Fatou分支有界性

讨论了几类用缺项幂级数表示的超越整函数Fatou集分支的有界性,并给出了其Fatou分支均有界的充分条件.     

An exact Fatou lemma for Gelfand integrals: a characterization of the Fatou property

We establish an exact version of Fatou’s lemma for Gelfand integrals of functions and multifunctions in dual Banach spaces without any order structure, and under the saturation property on the underlying measure space. The necessity and sufficiency of saturation for the Fatou property is demonstrated. Our result has a direct application to the equilibrium existence result for saturated economies without convexity assumptions.     

On a theorem of Fatou

We prove a result on the backward dynamics of a rational function nearby a point not contained in the -limit set of a recurrent critical point. As a corollary we show that a compact invariant subset of the Julia set, not containing critical or parabolic points, and not intersecting the -limit set ofrecurrent critical points, is expanding, thus extending a classical criteria of Fatou. We also prove that the boundary of a Siegel disk is always contained in the -limit set of arecurrent critical point.     

A general Fatou Lemma

A general Fatou Lemma is established for a sequence of Gelfand integrable functions from a vector Loeb space to the dual of a separable Banach space or, with a weaker assumption on the sequence, a Banach lattice. A corollary sharpens previous results in the finite-dimensional setting even for the case of scalar measures. Counterexamples are presented to show that the results obtained here are sharp in various aspects. Applications include systematic generalizations of the distribution of correspondences from the case of scalar Loeb spaces to the case of vector Loeb spaces and a proof of the existence of a pure strategy equilibrium in games with private and public information and with compact metric action spaces.     

On completely invariant Fatou components

Completely invariant components of the Fatou sets of meromorphic maps are discussed. Positive answers are given to Baker’s and Bergweiler’s problems that such components are the only Fatou components for certain classes of meromorphic maps. Both authors are supported by NSFC and the 973 Project.     

Fatou closedness under model uncertainty

We provide a characterization in terms of Fatou closedness for weakly closed monotone convex sets in the space of \({\mathcal P}\)-quasisure bounded random variables, where \({\mathcal P}\) is a (possibly non-dominated) class of probability measures. Applications of our results lie within robust versions the Fundamental Theorem of Asset Pricing or dual representation of convex risk measures.     

Fatou theorems for parabolic equations

For elliptic parabolic operators with time dependent coefficients, bounded and measurable, the absolute continuity of the two caloric measures plus a Fatou theorem are shown to hold on the parabolic boundary of a smooth cylinder given a Carleson-type condition on the coefficients of the operators, and assuming one of the measures is a center doubling measure. Given a stronger Carleson condition, and no doubling assumption, another kind of Fatou theorem result holds. The method of proof follows that of Fefferman, Kenig and Pipher.

     

Area of Fatou sets of trigonometric functions

We extend a result of McMullen to show that the area of the Fatou set of the sine function in a vertical strip of width is finite. This confirms a conjecture by Milnor.

     

On Fatou sets concerning Yang-Lee theory

The sets of the points corresponding to the complex phases of the Potts model on the diamond hierarchcal lattice are studied. These sets are the Fatou sets of a family of rational mappings. The topological structures of these sets are described completely.     

A Fatou theorem for F-harmonic functions

Mathematische Zeitschrift - In this paper we study a class of functions that appear naturally in some equidistribution problems and that we call F-harmonic. These are functions of the universal...     

On the structure of Fatou domains

Let U be a multiply-connected fixed attracting Fatou domain of a rational map f.We prove that there exist a rational map g and a completely invariant Fatou domain V of g such that(f,U) and(g,V) are holomorphically conjugate,and each non-trivial Julia component of g is a quasi-circle which bounds an eventually superattracting Fatou domain of g containing at most one postcritical point of g.Moreover,g is unique up to a holomorphic conjugation.     

On questions of Fatou and Eremenko

Let be a transcendental entire function and let be the set of points whose iterates under tend to infinity. We show that has at least one unbounded component. In the case that has a Baker wandering domain, we show that is a connected unbounded set.

     

The boundary of bounded polynomial Fatou components

We prove that, for a polynomial, every bounded Fatou component, with the exception of Siegel disks, has for boundary a Jordan curve. To cite this article: P. Roesch, Y. Yin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Quantitative Fatou Theorems and Uniform Rectifiability

We show that a suitable quantitative Fatou Theorem characterizes uniform rectifiability in the codimension 1 case.

     

A note on conditional Fatou Lemma

Summary In [4] Zheng Wei-an showed that Fatou's Lemma as presented in the book by Liptser and Shiryayev does not hold true. This paper presents a condition under which this Lemma does hold true.