Higher order local dimensions and Baire category

• Autorzy: Lars Olsen
• Języki publikacji: EN

Abstrakty

EN

Let X be a complete metric space and write 𝓟(X) for the family of all Borel probability measures on X. The local dimension $dim_{loc}(μ;x)$ of a measure μ ∈ 𝓟(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 ∈ 𝓟(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 ∈ 𝓟(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.

Kategorie tematyczne

• 28A80: Fractals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 204
• Numer: 1
• Strony: 1-20

Daty

wydano

2011

Twórcy

autor

Lars Olsen

• Department of Mathematics, University of St Andrews, St. Andrews, Fife KY16 9SS, Scotland

Identyfikatory

DOI

10.4064/sm204-1-1