The geometry of Whitney’s critical sets

In this paper, the complete geometric characterizations, including decomposition and compression theorems, are obtained for a connected and compact set to be a critical set in Whitney’s sense, i.e., a set such that there exists a differentiable function critical but not constant on it. The problem to characterize these sets geometrically was posed by H. Whitney [21] in 1935. We also provide a complete geometrical characterization for monotone Whitney arc, i.e., there exists a differentiable function critical but also increasing along the arc. All examples appearing in the literature are monotone Whitney arcs, for example, the examples by Whitney [21] and Besicovitch [2], Norton’s t-quasi-arcs with Hausdorff dimension > t [14], and self-similar arcs [19]. Furthermore, after introducing the notion of homogeneous Moran arc, we can completely characterize all the monotone Whitney arcs of criticality > 1, which include t-quasi arcs and self-conformal arcs. Some applications to arcs which are attractors of Iterated Function Systems are discussed, including self-conformal arcs, self-similar arcs and self-affine arcs. Finally, we give an example of critical arc such that any of its subarcs fails to be a t-quasi-arc for any t, providing an affirmative answer to an open question by Norton.