01-436 Francois Germinet, Serguei Tcheremchantsev

Generalized fractal dimensions of compactly supported measures on the negative axis (457K, pdf) Nov 27, 01

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Abstract. We study generalized fractal dimensions of measures, called the Hentschel-Procaccia dimensions and the generalized R\'enyi dimensions. We consider compactly supported Borel measures with finite total mass on a complete separable metric space. More precisely we discuss in great generality finiteness and equality of the different dimensions for negative values of their argument $q$. In particular we do not assume that the measure satisfies to the so called ``doubling condition". A key tool in our analysis is, given a measure $\mu$, the function $g(\eps)$, $\eps>0$, defined as the infimum over all points $x$ in the support of $\mu$ of the quantity $\mu(B(x,\eps))$, where $B(x,\eps)$ is the ball centered at $x$ and of radius $\eps$.

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