Dynamics in One Complex Variable: Introductory Lectures

These notes will study the dynamics of iterated holomorphic mappings from a Riemann surface to itself concentrating on the classical case of rational maps of the Riemann sphere They are based on introductory lectures given at Stony Brook during the Fall Term of 1989 90 and also in later years I am grateful to the audiences for a great deal of constructive criticism and to Branner Douady Hubbard and Shishikura who taught me most of what I know in this field Also I want to thank A Poirier S Zakeri and R Perez for their extremely helpful criticisms of various drafts There have been a number of extremely useful surveys of holomorphic dynamics over the years those of Brolin Douady Blanchard Lyubich Devaney Keen and Eremenko Lyubich as well as the textbooks by Bear don Steinmetz and Carleson Gamelin are particularly recommended to the reader Compare the list of references at the end and see Alexander for historical information This subject is large and rapidly growing These lectures are intended to introduce the reader to some key ideas in the field and to form a basis for further study The reader is assumed to be familiar with the rudiments of complex variable theory and of two dimensional differential geometry as well as some basic topics from topology The necessary material can be found for example in Ahlfors 1966 Hocking and Young Munkres and vVillmore This text studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself concentrating on the classical case of rational maps of the Riemann sphere The subject is large and rapidly growing These notes are intended to introduce the reader to some key ideas in the field and to form a basis for further study The reader is assumed to be familiar with the rudiments of complex variable theory and of two dimensional differential geometry as well as some basic topics from topology This volume studies the dynamics of iterated holomorphic mappings from a