Constructing Whitney sets via IFS with condensation

Abstract

In 1935, Whitney constructed a smooth function for which the Morse-Sard Theorem does not hold. Whitney's construction is closely related to certain compact connected set, which is called Whitney set now. From then on, there are a lot of works on Whitney sets. In this paper, we use IFS with condensation, a notion introduced by Barnsley and Demko in 1985, to construct Whitney arcs and Whitney sets. Our construction includes most early results as special cases.