Higher order local dimensions and Baire category

Lars Olsen. "Higher order local dimensions and Baire category." Studia Mathematica 204.1 (2011): 1-20. <http://eudml.org/doc/285427>.

@article{LarsOlsen2011,
abstract = {Let X be a complete metric space and write (X) for the family of all Borel probability measures on X. The local dimension $dim_\{loc\}(μ;x)$ of a measure μ ∈ (X) at a point x ∈ X is defined by $dim_\{loc\}(μ;x) = lim_\{r↘0\} (log μ(B(x,r)))/(log r)$ 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.},
author = {Lars Olsen},
journal = {Studia Mathematica},
keywords = {multifractals; local dimension; Baire category; averaging methods; summability method},
language = {eng},
number = {1},
pages = {1-20},
title = {Higher order local dimensions and Baire category},
url = {http://eudml.org/doc/285427},
volume = {204},
year = {2011},
}

TY - JOUR
AU - Lars Olsen
TI - Higher order local dimensions and Baire category
JO - Studia Mathematica
PY - 2011
VL - 204
IS - 1
SP - 1
EP - 20
AB - Let X be a complete metric space and write (X) for the family of all Borel probability measures on X. The local dimension $dim_{loc}(μ;x)$ of a measure μ ∈ (X) at a point x ∈ X is defined by $dim_{loc}(μ;x) = lim_{r↘0} (log μ(B(x,r)))/(log r)$ 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.
LA - eng
KW - multifractals; local dimension; Baire category; averaging methods; summability method
UR - http://eudml.org/doc/285427
ER -