Existence of Eigenvectors for Monotone Homogeneous Functions St

Abstra t. We onsider fun tions f : R n ! R n whi h are additively homogeneous and monotone in the produ t ordering on R n (topi al fun tions). We show that if some non-empty sub-eigenspa e of f is bounded in the Hilbert semi-norm then f has an additive eigenve tor and we give a Collatz-Wielandt hara terisation of the orresponding eigenvalue. The boundedness ondition is satis ed if a ertain dire ted graph asso iated to f is strongly onne ted. The Perron-Frobenius theorem for non-negative matri es, its analogue for the max-plus semiring, a version of the mean ergodi theorem for Markov hains and theorems of Bather and Zijm all follow as immediate orollaries.