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Estimates for the lower and the upper dimension of a measure. (Estimations de la dimension inférieure et de la dimension supérieure des mesures.) (French) Zbl 0903.28005

The function \(\tau\) which appears in multifractal analysis is known to be adapted to calculate the lower and the upper dimension of a measure, in particular when \(\tau'(1)\) does exist. The paper under review considers mainly the case where \(\tau\) has no derivative at 1. For that purpose, a notion of lower and upper dimension based on the Tricot dimension is introduced, which enables to interpret the left derivative of \(\tau\) at 1. Optimal estimates for the upper and the lower dimensions are thus obtained, for some good classes of measures, as well as upper and lower bounds in terms of Rényi dimensions. These results are applied to quasi-Bernoulli measures and yield the derivability of the function \(\tau\), precising results in [G. Brown, G. Michon and J. Peyrière, J. Stat. Phys. 66, 775-790 (1992)]. This nice paper ends with well-chosen examples and an application to quasi-symmetric functions.

MSC:

28A78 Hausdorff and packing measures
28A12 Contents, measures, outer measures, capacities
28D05 Measure-preserving transformations
28D20 Entropy and other invariants
60F10 Large deviations
28A80 Fractals
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