Discrete Holomorphic Local Dynamical Systems

Dynamics of Rational Surface Automorphisms
Eric Bedford
Abstract This is a 2-part introduction to the dynamics of rational surface
automorphisms. Such maps can be written in coordinates as rational functions
or polynomials. The first part concerns polynomial automorphisms of complex
2-space and includes the complex Henon family.
The second part concerns compact (complex) rational surfaces. The basic properties
of automorphisms of positive entropy are given, as well as the construction of
invariant currents and measures. This is illustrated by a number of examples.
1 Polynomial Automorphisms of C 2
1.1 Hénon Maps
A surface is said to be rational if it is birationally equivalent to the plane. The
purpose of these notes is to give an entry into the dynamics of the automorphisms
of rational surfaces. The first part is devoted to the complex Hénon family of
maps, which has been the most heavily studied family of invertible holomorphic
maps. Up to this point, the investigations of the Hénon maps have been guided
by the study of polynomial maps of one variable. The dynamics of polynomials
in the one-dimensional case has developed into a very rich topic, and these maps
are understood in considerable detail. Although the Hénon family is only partially
understood, its methods and results should provide motivation and guidance for the
understanding of other automorphisms in dimension 2. We have selected for discussion
only a part of what is known on the subject, and the reader is recommended to
consult the expository treatments in [MNTU] and[S], as well as the original works
[HOV1,2,FS1]andtheseriesofpapers[BS1,2,...],[BLS].
E. Bedford
Department of Mathematics, Indiana University, 47405 Bloomington, IN, USA
e-mail: bedford@indiana.edu
G. Gentili et al. (eds.), Holomorphic Dynamical Systems, 57
Lecture Notes in Mathematics 1998, DOI 10.1007/978-3-642-13171-4 2,
c○ Springer-Verlag Berlin Heidelberg 2010