Boundary Behaviour of Conformal Maps

Christian Pommerenke

• 07 december 2010
• 9783642081293

Samenvatting:

We study the boundary behaviour of a conformal map of the unit disk onto an arbitrary simply connected plane domain. Then the conformal map has many unexpected properties, for instance almost all the boundary is mapped onto almost nothing and vice versa.



There has been a great deal of recent interest in the boundary behaviour of conformal maps of the unit disk onto plane domains. In classical applications of conformal maps, the boundary tended to be smooth. This is not the case in many modern applications (e.g. for Julia sets). The first chapters present basic material and are also of interest for people who use conformal mapping as a tool. The later chapters deal in greater detail with classical material and, go into recent developments (e.g. by Makarov). The reader is assumed to know standard complex and real analysis. The subject of the book is developed from scratch except in a few places (e.g. quasiconformal maps) where there exist other very goodbooks: in such cases Pommerenke's emphasis is on giving additional information. There are over two hundred exercises most of which are easy and meant to test the reader's understanding of the text. Each chapter begins with an overview stating the main results informally