THE LIPSCHITZ STRUCTURE OF CONTINUOUS SELF-MAPS OF GENERIC COMPACT-SETS

Continuous self-maps of closed sets generic with respect to the Hausdo rff metric admit only a trivial Lipschitz structure, Unless f is the i dentity on some nonempty open set of E, the image of any set on which f is Lipschitz is nowhere dense in E. The set of points of differentia bility of f in E maps onto a first category subset of E. We apply thes e results and related ones to the study of omega-limit sets of continu ous functions. We show that while all nonvoid nowhere dense closed set s are omega-limit sets for continuous functions, most closed sets are not omega-limit sets for functions, most closed sets are not omega-lim it sets for functions exhibiting even minimal smoothness. (C) 1994 Aca demic Press, Inc.