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For a probability distribution $P$, Renyi fractal dimension is defined as

$$D_q = \lim_{\epsilon\rightarrow 0} \frac{R_q(P_\epsilon)}{\log(1/\epsilon)},$$ where $R_q$ is Renyi entropy of order $q$ and $P_\epsilon$ is the coarse-grained probability distribution (i.e. put in boxes of linear size $\epsilon$).

The question is if there are any phenomena, for which using non-trivial $q$ (i.e. $q\neq0,1,2,\infty$) is beneficial or naturally preferred?

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The Rényi entropy of order $q = \frac{1}{2}$ apperas in several measures of pure states entanglement, please see for example: Karol Zyczkowski, Ingemar Bengtsson: Relatively Pure states entanglement. This entropy has the property that for three state systems, the equientropy trajectories form circles with respect to the the Bhattacharyya distance, please see for example: Bengtsson Zyczkowski: Geometry of quantum states, page 57.

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Thank you for a nice example. However, $q=1/2$ is semi-trivial (as it is conjugated to $q=\infty$). –  Piotr Migdal Dec 16 '11 at 10:21

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