On strict Whitney arcs and $t$-quasi self-similar arcs

Abstract

A connected compact subset $E$ of $\mathbb{R}^{N}$ is said to be a strict Whitney set if there exists a real-valued $C^{1}$ function $f$ on $\mathbb{R}^{N}$ with $\nabla f|_{E}\equiv 0$ 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 $\mathbb{R}^{N}$ 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 $\Lambda $ to be a $t$-quasi-arc, and for the Hausdorff measure function on $\Lambda $ to be a strict Whitney function. We prove that if a regular self-similar arc has “minimal corner angle” $\theta_{\min }>0$, 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 $\tau $, 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 $t_{0}\ge 1$, there is a value of $\tau $ such that the corresponding self-similar arc is a $t$-quasi-arc for each $t>t_{0}$, but it is not a $t_{0}$-quasi-arc. For each $t_{0}>1$, there is a value of $\tau $ such that the corresponding self-similar arc is a $t_{0}$-quasi-arc, but it is a $t$-quasi-arc for no $t\in [1,t_{0})$.