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 $(X,\mathcal{F},\mu ,T)$ where the $\mu$-almost sure subadditivity condition $f_{n+m}\le f_n+f_m\circ T^n$ is relaxed to a $\mu$-almost sure, “gapped”, almost subadditivity condition of the form $f_{n+{\sigma }_m+m}\le f_n+{\rho }_n+f_m\circ T^{n+{\sigma }_n}$ for some nonnegative ${\rho }_n\in L^1(\mathrm{d}\mu )$ and ${\sigma }_n\in \mathbf{N}\cup \{0\}$ 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.