  A directed animal is a percolation cluster in the directed site percolation
model. The aim of this paper is to exhibit a strong relation between the
problem of computing the generating function $\G$ of directed animals on the
square lattice, counted according to the area and the perimeter, and the
problem of solving a system of quadratic equations involving unknown matrices.
We present some solid evidence that some infinite explicit matrices, the fixed
points of a rewriting like system are the natural solutions to this system of
equations: some strong evidence is given that the problem of finding $\G$
reduces to the problem of finding an eigenvector to an explicit infinite
matrix. Similar properties are shown for other combinatorial questions
concerning directed animals, and for different lattices.
