Abstract:
Tropical matrices are matrices with entries in $(\R\cup\{-\infty\}, \max,+)$, where $\max$ is seen as the “addition”. They appear as the limit of nonnegative matrices and in models from computer science/operations research, as they are strongly linked to weighted directed graphs. Their dynamical behavior is quite similar to nonnegative matrices, with a kind of Perron-Frobenius theorem but the limits are often reached, which allows some combinatorial studies. In this talk, I will present a common framework, known as topical maps, to deal with both nonnegative matrices and tropical ones. It is too general to study iterates of a given map but it is well suited to understand iterates of random maps, or equivalently products of random matrices. This is work in progress with Loïc Hervé (INSA Rennes).