THE EMBEDDING PROBLEM IN ITERATION THEORY

This is a survey paper on selected topics concerning the embeddability of given mappings in real ows. Particular attention will be paid to the relationship between the problem of the embed- dability and functional equations. Let X be a real manifold and f : X! X be a homeomorphism. A family of homeomorphismsff t : X ! X;t2 Rg such that f t f s = f t+s for t;s2 R and f 1 = f is said to be an embedding of f. The embedding is of class C r if for every x2 X the mapping t! f t (x) is continuous and all f t are of class C r . We concentrate on the cases where X is an open subset of R N ; X is a closed and an open interval, and X is a circle. We discuss the following problems: the existence of embeddings with suitable regularity; the conditions which imply the uniqueness of embeddings; the formulas expressing the above embeddings or their general constructions. R esum