 00341 J.M. Barbaroux, F. Germinet, S. Tcheremchantsev
 Generalized Fractal Dimensions:
Equivalences and Basic Properties
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Sep 6, 00

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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.
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