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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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