The theory of holomorphic dynamical systems is a subject of increasing
interest in mathematics, both for its challenging problems and for its
connections with other branches of pure and applied mathematics. A
holomorphic dynamical system is the datum of a complex variety and a
holomorphic object (such as a self-map or a vector ?eld) acting on it.
The study of a holomorphic dynamical system consists in describing the
asymptotic behavior of the system, associating it with some invariant
objects (easy to compute) which describe the dynamics and classify the
possible holomorphic dynamical systems supported by a given manifold.
The behavior of a holomorphic dynamical system is pretty much related to
the geometry of the ambient manifold (for instance, - perbolic manifolds
do no admit chaotic behavior, while projective manifolds have a variety
of different chaotic pictures). The techniques used to tackle such pr-
lems are of variouskinds: complexanalysis, methodsof real analysis,
pluripotential theory, algebraic geometry, differential geometry,
topology. To cover all the possible points of view of the subject in a
unique occasion has become almost impossible, and the CIME session in
Cetraro on Holomorphic Dynamical Systems was not an exception.