The topological structure of attractors for differentable functions

Abstract

Recent research has shown that there is a significant cleavage between the structure of \(\omega\)-limit sets for continuous functions, and the structure of \(\omega\)-limit sets for Lipschitz functions. The development of these results rested on measure theoretic considerations. In this paper we show that there is no such divergence when one considers the topological structure of these two classes of \(\omega\)-limit sets. We show that an every nowhere dense compact set is homeomorphic to an \(\omega\)-limit set for a differentiable function with bounded derivative.