The Lipschitz Structure of Continuous Self-Maps of Generic Compact Sets

Abstract

Continuous self-maps of closed sets generic with respect to the Hausdorff metric admit only a trivial Lipschitz structure. Unless ƒ is the identity on some nonempty open set of E, the image of any set on which ƒ is Lipschitz is nowhere dense in E. The set of points of differentiability of ƒ 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 ω-limit sets for continuous functions, most closed sets are not ω-limit sets for functions, most closed sets are not ω-limit sets for functions exhibiting even minimal smoothness.