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We extend some characterizations and inequalities for the eigenvalues ofnonnegative matrices, such as Donsker-Varadhan, Friedland-Karlin, Karlin-Ostinequalities, to nonnegative tensors. Our approach involves a correspondencebetween nonnegative tensors, ergodic control and entropy maximization: we showin particular that the logarithm of the spectral radius of a tensor is given byen entropy maximization problem over a space of occupation measures. We studyin particular the tropical analogue of the spectral radius, that wecharacterize as a limit of the classical spectral radius, and we give anexplicit combinatorial formula for this tropical spectral radius.

Spectral inequalities for nonnegative tensors and their tropical analogues