Semigroups of conformal mappings

Sets of conformal mappings distinguished by boundary conditions and forming semigroups with respect to the operation of composition are studied.
A description of one-parameter semigroups is given, and a connection between them and semiflows is established. With each semigroup there is associated an evolution equation that is an analogue of the Lëwner equation known in the theory of univalent functions. Existence theorems are obtained for the evolution equations, and their approximation properties are studied. It is also established that each mapping of the semigroups under consideration can be represented as a shift along solutions of the corresponding evolution equation.
Bibliography: 19 titles.