Optimal state points of the subadditive ergodic theorem

Let be an ergodic measure-preserving Borel measurable transformation of a separable metric space X that is not necessarily compact, and suppose that is a T-subadditive sequence of -measurable upper-bounded functions. In this paper, we prove that, if the sets of φn-discontinuities are of μ-measure 0 for all n ⩾ 1 and if the growth rates then *(x) < 0 for all points x in the basin BT(μ) of (T, μ). We apply this to considering the Oseledets regular points.