Francois Germinet, Serguei Tcheremchantsev
Generalized fractal dimensions of compactly supported measures on the 
negative axis
(457K, pdf)

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 $\eps$.