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On strict Whitney arcs and \(t\)-quasi self-similar arcs. (English) Zbl 1403.28008

The authors prove that a self-similar arc of Hausdorff dimension \(s > 1\) in \(\mathbb{R}^N\) is a strict Whitney set with criticality \(s\). Here, a strict Whitney set is a connected compact subset \(E\subset \mathbb{R}^N\) for which there exists a \(C^1\)-function \(f:\mathbb{R}^N\to \mathbb{R}\) with \(\nabla f |_E \equiv 0\) but \(f|_E\) is not constant.
In addition, so-called regular self-similar arcs are studied and necessary and sufficient conditions are derived for such an arc \(\Lambda\) to be a \(t\)-quasi-arc and for the Hausdorff measure function on \(\Lambda\) to be a strict Whitney function. Moreover, it is proved that if a regular self-similar arc has a strictly positive minimal corner angle then it is a \(t\)-quasi-arc and therefore its Hausdorff measure function on \(\Lambda\) a strict Whitney function.
An example of a one-parameter family of regular self-similar arcs with various features is also presented.

MSC:

28A80 Fractals
54F45 Dimension theory in general topology
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Full Text: arXiv Euclid

References:

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