  In the thermodynamic limit, a probabilistic cellular automaton can be
approximated by a deterministic nonlinear map. Here we construct such a map for
the forest fire problem. The construction is based on the results of the Monte
Carlo simulation, performed on a square lattice of million cells. The results
of the calculation are analyzed by means of the Hoshen--Kopelman algorithm
(HKA). The only parameter of the map describes the probability that a tree
appears at an empty cell during one time step. The obtained map seems to be
non-differentiable at the percolation threshold. The Lyapunov exponent for the
map is positive. Also, we found the cycle of length three by means of the
method of symbolic dynamics. The results are illustrated by the experimental
data on the forest fires in Canada in years 1970--2000. Although these data are
fortunately far from thermodynamic limit, their qualitative character is
reproduced for smaller lattices.
