The Berger-Wang formula for order-preserving homogeneous maps on cones

Abstract

We prove that the joint spectral radius and generalized spectral radius are equal for any bounded, equicontinuous family of order-preserving, homogeneous maps on a polyhedral cone. We also consider conditions which guarantee that the semigroup generated by a family of order-preserving, homogeneous maps is bounded when its generalized spectral radius $r(\mathcal{A}) = 1$. Finally, we extend the notions of joint and generalized spectral subradii to the setting of homogeneous maps on wedges.