2017/03/30 by Daowei Ma, Xin Wei, Ma, Daowei +3 · 1 citation
Mathematics · #28A80 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #General Topology (math.GN) #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1703.10665
openalex publication_date 2017/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 value 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).