A gapped generalization of Kingman's subadditive ergodic theorem

TL;DR

This paper generalizes Kingman's subadditive ergodic theorem by relaxing the subadditivity condition to a 'gapped' version, enabling new applications such as defining relative entropies for decoupled measures on shifts.

Contribution

It introduces a novel 'gapped' subadditivity condition and proves a corresponding ergodic theorem, extending the classical result to broader measure-preserving systems.

Findings

Established a generalized ergodic theorem under gapped subadditivity.

Applied the theorem to define relative entropies for decoupled measures.

Demonstrated the theorem's relevance to one-sided shift systems.

Abstract

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 $f_{n+m} \leq f_n + f_m \circ T^{n}$ is relaxed to a $\mu$-almost sure, "gapped", almost subadditivity condition of the form $f_{n+\sigma_m+m} \leq f_n +\rho_n + f_m \circ T^{n+\sigma_n}$ 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.