A gapped generalization of Kingman’s subadditive ergodic theorem

Abstract

We state and prove a generalization of Kingman’s ergodic theorem on a measure-preserving dynamical system (X,F,μ,T) where the μ-almost sure subadditivity condition fn+m ≤ fn + fm◦Tn is relaxed to a μ-almost sure, “gapped,” almost subadditivity condition of the form fn+σm+m≤fn+ρn+fm◦Tn+σn for some non-negative ρn ∈ L1(dμ) and σn∈N∪{0} that are suitably sublinear in n. This generalization has a first application to the existence of specific relative entropies for suitably decoupled measures on one-sided shifts.