00-341 J.-M. Barbaroux, F. Germinet, S. Tcheremchantsev

Generalized Fractal Dimensions: Equivalences and Basic Properties (580K, .ps) 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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