Trends and Problems in Complex Dynamics and Geometric Function Theory

Complex dynamical systems and nonlinear semigroup theory are not only of intrinsic interest, but are also important in the study of evolution problems. In recent years many developments have occurred, in particular, in the area of nonexpansive semigroups in Banach spaces. As a rule, such semigroups are generated by accretive operators and can be viewed as nonlinear analogs of the classical linear contraction semigroups. Another class of nonlinear semigroups consists of those semigroups generated by holomorphic mappings in complex finite and infinite dimensional spaces. Such semigroups appear in several diverse fields, including, for example, the theory of Markov stochastic branching processes, Krein spaces and the geometry of complex Banach spaces. In this talk based on the joint work with M.Elin and T.Sugawa we concentrate on trends and problems related to the nonlinear resolvent method and its connections to the classical geometric function theory. Also some applications to complex Banach algebras will be presented.