Ergodicity Conditions for Upper Transition Operators

We study ergodicity of bounded, sub-additive and non-negatively homogeneous maps on finite dimensional spaces which we call upper transition operators. We show that ergodicity coincides with the necessary and sufficient condition for a generalised Perron-Frobenius theorem for upper transition operators. We show that ergodicity is equivalent with regular absorningness of the upper transition operator: there has to be a top class that is regular and absorbing. Using this conditions, it can be shown that top class regularity can be checked by solving a linear eigenvalue problem where the stochastic matrix is build with the values of the upper transition operator for every atom. To check top class absorption it is shown that less than n evaluations of the upper transition operator have to be done.