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

We consider the iterates of bilinear functions over the semiring (max, +). Equivalently, our object of study can be viewed as recognizable tree series over the 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. Remark. Due to space limitations, the proofs have been omitted. A full version can be obtained from the authors on request.