On the Burnside problem for Semigroups of Matrices in the (max,+) Algebra

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\Gamma1g; 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. 1. Introduction The "(max,+)-algebra" is a traditional name for the semiring (R[f\Gamma1g; max; +), denoted Rmax in the sequel. This is a particular example of idempotent semiring (that is a semiring whose additive law satisfies a \Phi a = a), also known as dioid [17, 18, 2]. This algebraic structure has been popularized by its applications to Graph Theory and Operations Research [17, 8]. Linear operators in this algebra are central in Hamilton-Jacobi theory and in the study of exponential asymptotics [33]. The study of automata and semigroups of matrices over the analogous "tropical" semiring (N [ f+1g;min;+) has been ..