  We extend some characterizations and inequalities for the eigenvalues of
nonnegative matrices, such as Donsker-Varadhan, Friedland-Karlin, Karlin-Ost
inequalities, to nonnegative tensors. Our approach involves a correspondence
between nonnegative tensors, ergodic control and entropy maximization: we show
in particular that the logarithm of the spectral radius of a tensor is given by
en entropy maximization problem over a space of occupation measures. We study
in particular the tropical analogue of the spectral radius, that we
characterize as a limit of the classical spectral radius, and we give an
explicit combinatorial formula for this tropical spectral radius.
