Higher order local dimensions and Baire category
Tom 204 / 2011
Streszczenie
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.
