Mesures de Hausdorff et théorie de Perron-Frobenius des matrices non-négatives
Annales de l'Institut Fourier, Tome 35 (1985) no. 4, pp. 99-125.

Nous étudions des sous-ensembles parfaits de RN dont la structure dépend d’une matrice primitive à coefficients entiers ≥0. La dimension de Hausdorff d’un tel ensemble “fractal” s’exprime en fonction de la valeur propre réelle maximale de sa matrice associée. Nous utilisons le théorème de Perron-Frobenius pour calculer la valeur exacte (qui est finie et non-nulle) de la mesure de Hausdorff de cet ensemble, et nous montrons à quelle condition (géométrique) cette valeur est maximale.

Subsets of Rn whose structure depends on a non-negative primitive matrix with integer coefficients are studied. The Hausdroff dimension of such a “fractal” set is expressed in terms of the maximal real eigenvalue of its associated matrix. Using the Perron-Frobenius theorem, the Hausdorff measure (finite and non-zero) of the set is computed, and a (geometric) condition for this value to be maximal is proved.

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Marion, Jacques. Mesures de Hausdorff et théorie de Perron-Frobenius des matrices non-négatives. Annales de l'Institut Fourier, Tome 35 (1985) no. 4, pp. 99-125. doi : 10.5802/aif.1029. https://www.numdam.org/articles/10.5802/aif.1029/

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