Strict Whitney arcs and t-quasi self-similar arcs

Abstract

A connected compact subset E of RN is said to be a strict Whitney set if there exists a real-valued C1 function f on RN with ∇f |E ≡ 0 such that f is constant on no non-empty relatively open subsets of E. We prove that each self-similar arc of Hausdorff dimension s > 1 in RN is a strict Whitney set with criticality s. We also study a special kind of self-similar arcs, which we call “regular” self-similar arcs. We obtain necessary and sufficient conditions for a regular self-similar arc Λ to be a t-quasi-arc, and for the Hausdorff measure function on Λ to be a strict Whitney function. We prove that if a regular self-similar arc has “minimal corner angle” θmin > 0, then it is a 1-quasi-arc and hence its Hausdorff measure function is a strict Whitney function. We provide an example of a one-parameter family of regular self-similar arcs with various features. For some values of the parameter τ , the Hausdorff measure function of the self-similar arc is a strict Whitney function on the arc, and hence the self-similar arc is an s-quasi-arc, where s is the Hausdorff dimension of the arc. For each t0 ≥ 1, there is a value of τ such that the corresponding self-similar arc is a t-quasi-arc for each t > t0, but it is not a t0-quasi-arc. For each t0 > 1, there is a value of τ such that the corresponding self-similar arc is a t0-quasi-arc, but it is a t-quasi-arc for no t ∈ [1, t0).