On demontre un resultat dynamique sur l'iteration d'une application analytique complexe sur la sphere de Riemann

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Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains

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Aufgaben und Lehrsätze aus der Analysis

Recent results on the growth and distribution of the values of integral and meromorphic functions of genus zero

Harmonic measure and arclength

We study the dependence between various sets of limiting boundary values of an arbitrary function f of z. In particular we show that the Plesner theorem on angular boundary values of analytic functions generalizes in a natural way to arbitrary functions.

https://iopscience.iop.org/article/10.1070/IM1967v001n01ABEH000543/pdf

Boundary properties of arbitrary functions

(1) f (2) = iil[ (2 22,) VIZ I (written in the form f(x) = n (x -x,), when only real variables are under consideration). We are concerned primarily with the point set B = E (f) , defined as the set where the inequality 1 f (z)I < 1 is satisfied. In Section 2, we determine the infimum of the length of rhe longest interval in the se1 I? nI+ (L denotes the real axis) for the case where the X, lie on the interval I = I-1 , I] on L. Section 3 deals with the diameter of the set E nL, under the more general hypothesis that the x7, lie on the interval I, = [-I , Y] . In Section 4, we study the two-dimensional measure of the set E, under the restriction that all the 2, lie in the closute D of the unit disk D. We use a theorem of G. R. MacLane to show that the measure 1 B (f)I can be made arbitrarily small. We also deal briefly with the relation between the transfinite diameter of a closed set F and the infimum of II?(f) 1, under the hypothesis that all 2, lie in F. Section 5 is devoted to the problem of finding the greatest number of components thar the set E can have when the .z,) are required to lie in D; Section 4 deals with the sum of the diameters of the components of E, under the hypothesis that all z\\. lie in the disk D, : z i c; T. Section 7 concerns polynomials (1) for which the set B is connected. In Section 8, we consider two problems concerning the convexity of E and of the components of E, respectively. And in Section 9, we prove a theorem

Metric properties of polynomials

II. The Problem of Learning to Teach

Sets of convergence of Taylor series. II.

Two monotonic, singular, uniformly almost smooth functions

1. Prime ends. The foundation for the study of boundaries of simply connected domains in the plane was laid by Carathéodory [2], who defined ends in general and prime ends in particular, and who classified prime ends into four kinds. To facilitate a brief survey of the subject of prime ends, I introduce a few definitions concerning a simply connected domain B. In my definitions, I follow essentially the work of Carathéodory, except that, for the sake of brevity, I omit the description of ends in general and aim directly at prime ends. A sequence K of crosscuts cn of B is a chain provided (i) the diameter of cn tends to 0 as »—» <x> ; (ii) for each index nf the set cnr\cn+i (where the bar indicates closure) is empty; (iii) some fixed point 0 in B cannot be joined to any crosscut cn (n> 1) by any path in B which does not meet the crosscut cw-i. Two chains K = {cn} and K! = {cn'} in B are equivalent provided each crosscut cn effects a separation, relative to B, of the point 0 from all except finitely many of the crosscuts cn'. An equivalence class of chains in B is a prime end of B. If P is a prime end of Bt let K~ {cn} denote a chain which belongs to P , and for each index n let Bn denote that subdomain of B which is determined by cn and does not contain the point 0. The set I(P) = f\Bn will be called the impression of P. It should be remarked that Carathéodory and some other writers applied the term prime end to the point set I(P)f but that they regarded as distinct two prime ends Pi and P* corresponding to two nonequivalent chains, even in cases where the two sets /(Pi) and I(P*) are identical. The distinction between a prime end and its impression formalizes the ideas which are involved. Carathéodory's principal theorem on the correspondence between boundaries under conformai mappings [2, p. 350] can be expressed as follows: Iff(z) maps the unit disk conformally and one-to-one onto the

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George Piranian

Papers

Metric properties of polynomials

II. The Problem of Learning to Teach

Sets of convergence of Taylor series. II.

Two monotonic, singular, uniformly almost smooth functions

The boundary of a simply connected domain