Strong and weak duality for fractal dimension in Euclidean space.(English)Zbl 0866.28004
Let \(X\) be a metric space, \(\mu\) a finite Borel measure on \(X\) and \(\alpha^-_\mu\), \(\alpha^+_\mu\) the lower, resp. upper pointwise dimension maps defined on \(X\). If \(A\subset \mathbb{R}^d\) and \({\mathcal P}(A)\) the probability measures on \(A\) then the author proves two representations of the Hausdorff and packing dimension \(\dim A\), resp. \(\text{Dim }A\) in terms of \(\alpha^-_\mu\) and \(\alpha^+_\mu\). The first one is called the weak duality principle and the second one, which holds for non-empty analytic sets \(A\), the strong duality principle. These results appear as corollaries to the extended Frostman resp. antiFrostman lemmas proved by the author. The main difference to the traditional Frostman lemma is the use of dyadic cubes instead of balls. Finally, the connection to the information dimension and the variational principle due to the author and Olsen is established.
Reviewer: H.Haase (Greifswald)
Keywords:
Hausdorff dimension;extended Frostman lemma;pointwise dimension maps;packing dimension;weak duality principle;strong duality principleCite
Full Text:DOI
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