J.-M. Barbaroux, F. Germinet, S. Tcheremchantsev
Generalized Fractal Dimensions: 
Equivalences and Basic Properties
(580K, .ps)

ABSTRACT.  Given a positive probability Borel measure $\mu$, we establish some 
basic properties of the associated functions $\tau_\mu^\pm(q)$ and of 
the generalized fractal dimensions $D_\mu^\pm(q)$ for $q\in\R$. We 
first give the connections between the generalized fractal dimensions, 
the R\'enyi dimensions and the mean-$q$ dimensions when $q>0$. We then 
use these relations to prove some regularity properties for 
$\tau_\mu^\pm(q)$ and $D_\mu^\pm(q)$; we also give some estimates for 
these functions as well as for their product of convolution. We finally 
present some calculations for specific examples.