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Higher order local dimensions and Baire category

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Olsen L.

Studia Mathematica

2011

tom 204

nr 1

1-20

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.

Ograniczanie wyników

1 Studia Mathematica

1 Olsen L.

1 2011

Wyniki wyszukiwania

Higher order local dimensions and Baire category