Higher order local dimensions and Baire category

Lars Olsen¹

¹ Department of Mathematics University of St Andrews St. Andrews, Fife KY16 9SS, Scotland

Studia Mathematica, Tome 204 (2011) no. 1, pp. 1-20

Résumé

Let $X$ be a complete metric space and write $\mathcal{P}(X)$ for the family of all Borel probability measures on $X$. The local dimension $\dim_{\mathsf{loc}}(\mu;x)$ of a measure $\mu\in\mathcal{P}(X)$ at a point $x\in X$ is defined by $$ \dim_{\mathsf{loc}}(\mu;x) = \lim_{r\searrow0}\frac{\log\mu(B(x,r))}{\log r} $$ whenever the limit exists, and plays a fundamental role in multifractal analysis. It is known that if a measure $\mu\in\mathcal{P}(X)$ satisfies a few general conditions, then the local dimension of $\mu$ exists and is equal to a constant for $\mu$-a.a. $x\in X$. In view of this, it is natural to expect that for a fixed $x\in X$, 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 $\mu\in\mathcal{P}(X)$ is so extremely irregular that, for a fixed $x\in X$, the local dimension function, $$ r \mapsto \frac{\log\mu(B(x,r))}{\log r}, $$ of $\mu$ at $x$ remains divergent as $r\searrow 0$ 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.

Détail

DOI : 10.4064/sm204-1-1

Keywords: complete metric space write mathcal family borel probability measures local dimension dim mathsf loc measure mathcal point defined dim mathsf loc lim searrow frac log log whenever limit exists plays fundamental role multifractal analysis known measure mathcal satisfies few general conditions local dimension exists equal constant mu a view natural expect fixed local dimension typical sense baire category measure exists quite surprisingly prove local dimension typical measure fails exist spectacular namely behaviour typical measure mathcal extremely irregular fixed local dimension function mapsto frac log log remains divergent searrow even after being averaged smoothened out general powerful averaging methods including example higher order riesz hardy logarithmic averages ces averages