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

Heurteaux, Yanick

Ann. Inst. Henri Poincaré, Probab. Stat. 34, No. 3, 309-338 (1998).

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.

Reviewer: V.Berthé (Marseille)

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

Keywords:

Hausdorff measure; packing measure; Hausdorff dimension; lower and upper dimension; Cantor sets; multifractal analysis; Tricot dimension; Rényi dimensions; quasi-Bernoulli measures