Evolution families of conformal mappings with fixed points and Loewner-Kufarev equation

Evolution families of conformal mappings of the unit disk into itself, having interior and boundary fixed points, are studied. Conditions of differentiability of evolutionary families and an existence theorem and uniqueness for the evolution equation are received. The convergence theorem is obtained which gives a description of topology of locally uniform convergence of evolution families in terms of infinitesimal generating functions. The basic result is made by an embedding theorem according to which any conformal mapping of the unit disk into itself with two fixed points can be embedded into a differentiable evolution family of such maps. The theorem allows to extend abilities of the parametrical method of the univalent function theory. On this way the problem about a mutual variation of the derivative in an interior point and the angular derivative in a boundary fixed point for a class of maps of an unit disk in itself is solved. In particular, the rotation theorem in this class of maps is received.