Bilinear functions and trees over the (max,+) semiring

International audienceWe consider the iterates of bilinear functions over the semiring max-plus. Equivalently, our object of study can be viewed as recognizable tree series over the max-plus semiring max. In this semiring, a fundamental result associates the asymptotic behaviour of the iterates of a linear function with the maximal average weight of the circuits in a graph naturally associated with the function. Here we provide an analog for the `iterates' of bilinear functions. We also give a triple recognizing the formal power series of the worst case behaviour