A gapped generalization of Kingman’s subadditive ergodic theorem

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 f n + σ m + m ≤ f n + ρ n + f m ◦ T n + σ n for some non-negative ρn ∈ L¹(dμ) and σ n ∈ 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.

Duke Scholars

Author Renaud Raquépas Mathematics