Some generic properties of concentration dimension of measure. (English) Zbl 1177.28014

Myjak, Józef; Szarek, Tomasz

Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 6, No. 1, 211-219 (2003).

Summary: Let \(K\) be a compact quasi self-similar set in a complete metric space \(X\) and let \(\mathfrak{M}_{1} (K)\) denote the space of all probability measures on \(K\), endowed with the Fortet–Mourier metric. We will show that for a typical (in the sense of Baire category) measure in \(\mathfrak{M}_{1} (K)\) the lower concentration dimension is equal to the Hausdorff dimension of \(K\).

MSC:

28A78	Hausdorff and packing measures
54E45	Compact (locally compact) metric spaces
37B99	Topological dynamics