  We consider convex maps f:R^n -> R^n that are monotone (i.e., that preserve
the product ordering of R^n), and nonexpansive for the sup-norm. This includes
convex monotone maps that are additively homogeneous (i.e., that commute with
the addition of constants). We show that the fixed point set of f, when it is
non-empty, is isomorphic to a convex inf-subsemilattice of R^n, whose dimension
is at most equal to the number of strongly connected components of a critical
graph defined from the tangent affine maps of f. This yields in particular an
uniqueness result for the bias vector of ergodic control problems. This
generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer
and Federgruen, for ergodic control problems with finite state and action
spaces, which correspond to the special case of piecewise affine maps f. We
also show that the length of periodic orbits of f is bounded by the cyclicity
of its critical graph, which implies that the possible orbit lengths of f are
exactly the orders of elements of the symmetric group on n letters.
