Mesures de Hausdorff et théorie de Perron-Frobenius des matrices non-négatives

Abstract

Subsets of R n 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.