Existence of eigenvectors for monotone homogeneous functions

We consider functions f: Rⁿ → Rⁿwhich are additively homogeneous and monotone in the product ordering on Rⁿ (topical functions). We show that if some non-empty sub-eigenspace of f is bounded in the Hilbert semi-norm then f has an additive eigenvector and we give a Collatz-Wielandt characterisation of the corresponding eigenvalue. The boundedness condition is satisfied if a certain directed graph associated to f is strongly connected. The Perron-Frobenius theorem for non-negative matrices, its analogue for the max-plus semiring, a version of the mean ergodic theorem for Markov chains and theorems of Bather and Zijm all follow as immediate corollaries.