Fractal boundaries are not typical
Let \(M\subset \mathbb{R}^{n}\) be a \(C^{1}\) compact \(n\)-dimensional manifold with boundary. For a \(C^{1}\) embedding \(f\in \partial M\times\left[0,\infty\right)\rightarrow M \) such that \(f((x,0))=x\) for \(x\in\partial M\), the author defines the space \({\mathcal{F}}_{f}(M)\), the frame-approximation space of \(M\), the elements of which are obtained by deforming \(\partial M .\) The elements of \({\mathcal{F}}_{f}(M)\) are defined to be sets \(\{f(x,g(x))\in\mathbb{R}^{n}, x\in\partial M\}\), where \(g\in C^{0}(\partial M,\left[0,\infty\right)).\) The space \({\mathcal{F}}_{f}(M)\) endowed with the Hausdorff metric is a complete metric space. It is proved that a typical element of \({\mathcal{F}}_{f}(M)\) is nowhere-differentiable and has Hausdorff dimension equal to \(n-1\).
scientific article; zbMATH DE number 5121445