David Shoikhet: Old and New in Complex Dynamical Systems

Historically, complex dynamics and geometrical function theory have been intensively developed from the beginning of the twentieth century. They provide the foundations for broad areas of mathematics. In the last Öfty years the theory of holomorphic mappings on complex spaces has been studied by many mathematicians with many applications to
nonlin- ear analysis, functional analysis, di§erential equations, classical and quan- tum mechanics. The laws of dynamics are usually presented as equations of motion which are written in the abstract form of a dynamical system: dx/dt + f(x) = 0, where x is a variable describing the state of the system dt under study, and f is a vector-function of x. The study of such systems when f is a monotone or an accretive (generally nonlinear) operator on the underlying space has been recently the subject of much research by
an- alysts working on quite a variety of interesting topics, including boundary value problems, integral equations and evolution problems .There is a long history associated with the problem on iterating holomorphic mappings and their Öxed points, the work of G. Julia, J.
Wol§ and C. CarathÈodory being among the most important.

In this talk we give a brief description of the classical statements which combine celebrated Juliaís Theorem in 1920 , CarathÈodoryís contribution in 1929 and Wol§ís boundary version of the Schwarz Lemma in
1926 and their modern interpretations for discrete and continuous semigroups.

We study the asymptotic behavior of one-parameter continuous semi- groups (áows) of holomorphic mappings and present angular character- istics of the áows trajectories at their Denjoy-Wol§ points, as well as at their regular repelling points (whenever they exist). This enables us by using linearization models in the spirit of functional Schroederís and Abelís equations to establish new rigidity properties of holomorphic
gen- erators which cover the famous Burns-Krantz Theorem and to solve a Nevanlinna-Pick type boundary interpolation problem for generators.