Let X be a complete metric space and write (X) for the family of all Borel probability measures on X. The local dimension of a measure μ ∈ (X) at a point x ∈ X is defined by whenever the limit exists, and plays a fundamental role in multifractal analysis. It is known that if a measure μ ∈ (X) satisfies a few general conditions, then the local dimension of μ exists and is equal to a constant for μ-a.a. x ∈ X. In view of this, it is natural to expect that for a fixed x ∈ X, the local dimension of a typical (in the sense of Baire category) measure exists at x. Quite surprisingly, we prove that this is not the case. In fact, we show that the local dimension of a typical measure fails to exist in a very spectacular way. Namely, the behaviour of a typical measure μ ∈ (X) is so extremely irregular that, for a fixed x ∈ X, the local dimension function, r ↦ (log μ(B(x,r)))/(log r), of μ at x remains divergent as r ↘ 0 even after being “averaged” or “smoothened out” by very general and powerful averaging methods, including, for example, higher order Riesz-Hardy logarithmic averages and Cesàro averages.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-1-1,
author = {Lars Olsen},
title = {Higher order local dimensions and Baire category},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {1-20},
zbl = {1225.28007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-1-1}
}
Lars Olsen. Higher order local dimensions and Baire category. Studia Mathematica, Tome 204 (2011) pp. 1-20. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-1-1/