  The aim of this manuscript is to understand the dynamics of matrix products
in a max algebra. A consequence of the Perron-Fr\"{o}benius theorem on periodic
points of a nonnegative matrix is generalized to a max algebra setting. The
same is then studied for a finite product associated to a $p$-lettered word on
$N$ letters arising from a finite collection of nonnegative matrices, with each
member having its maximum circuit geometric mean at most 1.
