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On pointwise dimensions and spectra of measures. (English. Abridged French version) Zbl 1012.37017
In the paper the generalization of Pesin’s Carathéodory construction is considered. Let $$X$$ be a measurable subset of $$\mathbb{R}^N$$, $$\tau$$ a real-valued set function, $$Q\subset (0,\infty)$$ a countable set with $$0$$ in its closure. $$\alpha_\tau(A,q)$$, called the spectrum for the set-function $$\tau$$ of the set $$A$$, is defined for all $$A\subset X$$. Following Pesin the authors introduce the spectrum of the Borel probability measure on $$X$$ as $$\alpha^\mu_\tau(q):= \inf\{\alpha_\tau(Y, q)$$: $$Y$$ measurable $$\subset X$$, $$\mu(Y)= 1\}$$.
Then the authors introduce a new “local version” of the preceding quantity, called the lower $$q$$-pointwise dimension of $$\mu$$ at the point $$x$$, and define by $d^\tau_{\mu,q}(x):= \liminf_{Q\ni\varepsilon\to 0} \inf_{y\in B(x,\varepsilon)} {\log\mu(B(y, \varepsilon))+ q\tau(B(y,\varepsilon))\over \log\varepsilon},$ where $$B(y,\varepsilon)$$ is the ball of the radius $$\varepsilon$$ and with the center in $$y$$. The main result of the paper: For every probability measure $$\mu$$ on $$X$$ and for all $$q\in \mathbb{R}$$ $\alpha^\mu_\varepsilon(q)= \text{ess sup }\alpha\tau_{\mu,q}.$ There is an example: dimension characteristics for Poincaré recurrence.

##### MSC:
 37C45 Dimension theory of smooth dynamical systems 28A78 Hausdorff and packing measures 28A80 Fractals
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