 01436 Francois Germinet, Serguei Tcheremchantsev
 Generalized fractal dimensions of compactly supported measures on the
negative axis
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Nov 27, 01

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Abstract. We study generalized fractal dimensions of measures, called the HentschelProcaccia
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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