On Semigroups of Matrices in the (max,+) Algebra
نویسنده
چکیده
We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max;+)-algebra (that is, the semiring (R[ f 1g;max;+)), and also for semigroups of (max;+)-linear projective maps with rational entries. An application to the estimation of the Lyapunov exponent of certain products of random matrices is also discussed. Key-words: Semigroups, Burnside Problem, (max;+) algebra, Lyapunov Exponents, Projective Space. (Résumé : tsvp) e-mail: [email protected], tel: (33 1) 39.63.52.58 Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Téléphone : (33 1) 39 63 55 11 – Télécopie : (33 1) 39 63 53 30 Sur les semigroupes de matrices dans l’algèbre (max,+) Résumé : Nous montrons que la réponse au problème de Burnside est positive pour les semigroupes de matrices à coefficients dans “l’algèbre (max;+)” (c’est-à-dire le semianneau (R[ f 1g;max;+)) ainsi que pour les semigroupes d’applications linéaires projectives à coefficients rationnels dans la même algèbre. On donne une application à l’estimation de l’exposant de Lyapunov de certains produits (max;+) de matrices aléatoires. Mots-clé : Semigroupes, Problème de Burnside, algèbre (max;+), Exposants de Lyapunov, Espace projectif. On Semigroups of Matrices in the (max,+) Algebra 1
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