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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.

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