Expansions of Lyapunov Exponentsand Forgetful

We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memory-loss property which holds generically for random products of matrices over the max-plus semiring, and in particular, for Tetris-like heaps of pieces models, we give a series expansion formula for the Lyapunov exponent, as a function of the probability law. In the case of rational probability laws, we show that the Lya-punov exponent is an analytic function of the parameters of the law, in a domain that contains the absolute convergence domain of a partition function associated to a special \forgetful" monoid, deened by generators and relations.