The embedding problem in iteration theory

• Main Author: Zdun Marek Cezary
• Format: Article
• Language: English
• Published: EDP Sciences 2014-11-01
• Series: ESAIM: Proceedings and Surveys
• Subjects: iteration groups; flows; embedding; functional equations; set-valued functions
• Online Access: http://dx.doi.org/10.1051/proc/201446008

This is a survey paper on selected topics concerning the embeddability of given mappings in real flows. Particular attention will be paid to the relationship between the problem of the embeddability and functional equations.Let X be a real manifold and f:X → X be a homeomorphism. A family of homeomorphisms {ft:X → X,t ∈ R} such thatft°fs = ft + s for t,s ∈ R and f1 = f is said to be an embedding of f. The embedding is of class Cr if for every x ∈ X the mapping t → ft(x) is continuous and all ft are of class Cr. We concentrate on the cases where X is an open subset of RN,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.