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

[摘要] Continuous self-maps of closed sets generic with respect to the Hausdorff metric admit only a trivial Lipschitz structure, Unless f is the identity 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 differentiability of f in E maps onto a first category subset of E. We apply these results and related ones to the study of omega-limit sets of continuous functions. We show that while all nonvoid nowhere dense closed sets are omega-limit sets for continuous functions, most closed sets are not omega-limit sets for functions, most closed sets are not omega-limit sets for functions exhibiting even minimal smoothness. (C) 1994 Academic Press, Inc.

[发布日期] 1994-12-15