Holomorphic Dynamics and Renormalization

The papers collected in this volume reflect some of the directions of research in two closely related fields: Complex Dynamics and Renormalization in Dynamical Systems. While dynamics of polynomial mappings, particularly quadratics, has by now reached a mature state of development, much less is known about non-polynomial rational maps. The reader will be introduced into this fascinating world and a related area of transcendental dynamics by the papers in this volume. A graduate student will find an area rich with open problems and beautiful computer simulations. A survey by V. Nekrashevych introduces the reader to iterated monodromy groups of rational mappings, a recently developed subject that links geometric group theory to combinatorics of rational maps.In this new language, many questions related to Thurston's theory of branched coverings of the sphere can be answered explicitly. Renormalization theory occupies a central place in modern Complex Dynamics. The progress in understanding the structure of the Mandelbrot set, polynomial Julia sets, and Feigenbaum-type universalities stems from renormalization techniques. Renormalization of circle maps and rotation domains, such as Siegel disks, can be understood in the context of the classical KAM theory. Corresponding phenomena in higher dimensions, such as universal scaling in area-preserving maps in 2D, on the boundary of KAM, pose a challenging problem. A survey by H. Koch and several other papers in the volume will introduce the reader to this direction of study.