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Generalized fractal dimensions: equivalences and basic properties. (English) Zbl 1050.28006
Given a positive probability Borel measure $$\mu$$, define for any $$q\in {\mathbf R}$$ and $$\varepsilon \in (0,1)$$, the integral $$I_\mu(q, \varepsilon) =\int_{\text{supp }\mu} \mu([x-\varepsilon, x+\varepsilon])^{q-1} d\mu(x)$$ and the functions $\tau_\mu^-(q) =\liminf_{s\downarrow 0}\frac{\log I_\mu(q,\varepsilon)}{-\log \varepsilon}, \quad \quad \tau_\mu^+(q) =\limsup_{s\downarrow 0}\frac{\log I_\mu(q,\varepsilon)}{-\log \varepsilon}.$ The lower and upper generalized fractal dimensions of $$\mu$$ are, respectively, defined for $$q\in {\mathbf R}\setminus \{1\}$$ as $D_\mu^-(q) =\liminf_{s\downarrow 0}\frac{\log I_\mu(q,\varepsilon)}{(q-1)\log \varepsilon}, \qquad D_\mu^+(q) =\limsup_{s\downarrow 0}\frac{\log I_\mu(q,\varepsilon)}{(q-1)\log \varepsilon}.$ The entropy dimensions are $D_\mu^-(1) =\liminf_{s\downarrow 0}\frac{\int_{\text{supp\,}\mu} \log(\mu[x-\varepsilon, x+\varepsilon]) \,d\mu(x)}{\log \varepsilon},$ $D_\mu^+(1) =\limsup_{s\downarrow 0}\frac{\int_{\text{supp\,}\mu} \log(\mu[x-\varepsilon, x+\varepsilon])\, d\mu(x)}{\log \varepsilon}.$ Note that in the literature the above dimensions are often referred to as Hentschel-Procaccia generalized dimensions. In the present article the authors establish some basic properties of $$\tau_\mu^{\pm}(q)$$ and $$D_\mu^{\pm}(q)$$ for $$q\in {\mathbf R}$$. They first give the equivalence of the Hentschel-Procaccia dimensions with the Rényi dimensions and the mean-$$q$$ dimensions for $$q>0$$. Then they use these relations to study for which set of $$q'$$s the functions $$\tau_\mu^{\pm}(q)$$, $$D_\mu^{\pm}(q)$$ are finite and $$D_\mu^{\pm}(q)\in [0,1]$$ and further prove some regularity properties for $$\tau_\mu^{\pm}(q)$$ and $$D_\mu^{\pm}(q)$$. They also give some estimates for these functions, in particular estimates on their behavior at $$\pm\infty$$, as well as for the dimensions corresponding to convolution of two measures. Finally, as an illustration of their behavior, the authors provide calculations of the generalized fractal dimension $$D_\mu^{\pm}(q)$$ for some specific examples.

##### MSC:
 28A80 Fractals 28A78 Hausdorff and packing measures
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