On subsets of Hilbert space having finite Hausdorff dimension. (Russian) Zbl 0652.46016

Shubov, V. I.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 163, 154-165 (1987).

Let \(X_ 1\), \(X_ 2\) be two Hilbert spaces such that \(X_ 2\) is densely and compactly included in \(X_ 1\). Next let \({\mathcal M}\) be a subset of \(X_ 2\) and \(\dim_ H^{(i)}{\mathcal M}\) the Hausdorff measure of \({\mathcal M}\) with respect to the metric in \(X_ i\) \((i=1,2)\). The author is interested in the problem whether there exists a subset \({\mathcal M}\) of \(X_ 2\) satisfying \(\dim_ H^{(1)}{\mathcal M}<\infty\) and \(\dim_ H^{(2)}{\mathcal M}=\infty\) or not. This problem is derived from the theory of attractors for certain nonlinear evolution equations. Let \(\{\lambda^ 2_ k\}\) be the increasing sequence of positive numbers satisfying \(\lim_{k\to \infty}\lambda^ 2_ k=\infty\), and \(\{q_ k\}\) the increasing sequence of positive integers. The pair \(X_ 1=\ell_ 2\) and \(X_ 2=\{x=(x_ 1,x_ 2,...)\in \ell_ 2\); \(\| x\|_ 2=(\sum^{\infty}_{k=1}\lambda^ 2_ kx^ 2_ k)^{1/2}\}\) is treated here. As the subset \({\mathcal M}\) he takes the sets contained in \(\{x\in \ell_ 2\); \(x_ q=0\) for \(q\neq q_ k\}\) \((\subset X_ 2)\). He gives the lower bounds of \(\lambda^ 2_{q_ k}\) and the restrictions on \(x_{q_ k}\), and he derives the following two examples as an affirmative answer to the problem: The first example shows a bounded subset \({\mathcal M}\) in \(x_ 2\) with finite \(\dim_ H^{(1)}{\mathcal M}\) such that \({\mathcal M}\) cannot be covered by any family of countable compact sets in \(X_ 2\). Next let N(\({\mathcal M},\epsilon)\) be the number of elements in minimal covering of \({\mathcal M}\) by closed spheres in \(X_ 2\) with radius \(\epsilon\), and \(h^{(2)}({\mathcal M})\equiv \overline{\lim}_{\epsilon \to 0}\{\ell n N({\mathcal M},\epsilon)\}/\{\ell n(1/\epsilon)\}\). \(h^{(2)}({\mathcal M})\) is not less than \(\dim_ H^{(2)}({\mathcal M})\) and called informational measure. The second example shows a compact subset \({\mathcal M}\) in \(X_ 2\) with finite \(\dim_ H^{(1)}{\mathcal M}\) and with infinite \(h^{(2)}({\mathcal M})\).

Reviewer: H.Yamagata

MSC:

46C99	Inner product spaces and their generalizations, Hilbert spaces
28A75	Length, area, volume, other geometric measure theory
37C70	Attractors and repellers of smooth dynamical systems and their topological structure
94A17	Measures of information, entropy

Keywords:

Hausdorff measure; theory of attractors for certain nonlinear evolution equations; informational measure