Skip to main content
main-content

Über dieses Buch

Historically, complex analysis and geometrical function theory have been inten­ sively developed from the beginning of the twentieth century. They provide the foundations for broad areas of mathematics. In the last fifty years the theory of holomorphic mappings on complex spaces has been studied by many mathemati­ cians with many applications to nonlinear analysis, functional analysis, differential equations, classical and quantum mechanics. The laws of dynamics are usually presented as equations of motion which are written in the abstract form of a dy­ namical system: dx / dt + f ( x) = 0, where x is a variable describing the state of the system under study, and f is a vector function of x. The study of such systems when f is a monotone or an accretive (generally nonlinear) operator on the under­ lying space has been recently the subject of much research by analysts working on quite a variety of interesting topics, including boundary value problems, integral equations and evolution problems (see, for example, [19, 13] and [29]). In a parallel development (and even earlier) the generation theory of one­ parameter semigroups of holomorphic mappings in en has been the topic of interest in the theory of Markov stochastic processes and, in particular, in the theory of branching processes (see, for example, [63, 127, 48] and [69]).

Inhaltsverzeichnis

Frontmatter

Preliminaries

Abstract
In this short chapter we compile the series of very basic notions and results, probably familiar to most readers. As we mentioned in the Preface, only a very modest preliminary knowledge is required to read the following material. Nevertheless, certain fundamental topics, such as those related to integral representations, convergence theorems, and fixed point principles, will be used throughout the text, and therefore should be presented at least an auxiliary material.
David Shoikhet

Chapter 1. The Wolff-Denjoy theory on the unit disk

Abstract
There is a long history associated with the problem on iterating holomorphic mappings and their fixed points, the work of H.A. Schwarz (1869), G. Pick (1916), G. Julia (1920), J. Wolff (1926), A. Denjoy (1926) and C. Carathéodory (1929) being among the most important.
David Shoikhet

Chapter 2. Hyperbolic geometry on the unit disk and fixed points

Abstract
Another look at the Wolff-Denjoy theory is the using of the so called hyperbolic metric of a domain.
David Shoikhet

Chapter 3. Generation theory on the unit disk

Abstract
For physical, chemical, and biological applications it is sometimes preferable to study a great variety of iterative processes, including the processes of continuous time. In spite of their simplicity these processes have been applicable in many fields, involving mathematical areas such as geometry, theory of stochastic branching processes, operator theory on Hardy spaces, and optimizations methods. A problem that has interested mathematicians since the time of Abel is how to define n-th iterate of function when n is not integer.
David Shoikhet

Chapter 4. Asymptotic behavior of continuous flows

Abstract
In this chapter we want to trace a connection of the iterating theory of functions in one complex variable and the asymptotic behavior of solutions of ordinary differential equations governed by evolution problems. Therefore our terminology is related to both these topics.
David Shoikhet

Chapter 5. Dynamical approach to starlike and spirallike functions

Abstract
This chapter is devoted to showing some relationships between semigroups and the geometry of domains in the complex plane. Mostly we will study those univalent (one-to-one correspondence) functions on the unit disk whose images are starshaped or spiralshaped domains. Several important aspects, however, had to be omitted, e.g. convex and close-to-convex functions (see, for example, [57, 55]), and other different classes of univalent functions. We have selected the forthcoming material according to the guiding principle that the demonstrated methods may be generalized to higher dimensions. For example, the celebrated Koebe One Quarter Theorem states that the image of a univalent function h on ∆ normalized by the condition h(0) = 0 and h’(0) = 1 contains a disk of radius 1/4. This theorem is no longer true at higher dimensions. Nevertheless, the dynamical approach analogues of the Koebe theorem have been recently established and used for subclasses of starlike (or spirallike) functions (see, for example [141, 109, 26, 56, 14]).
David Shoikhet

Backmatter

Weitere Informationen