# Ratio ergodic theorems: From Hopf to Birkhoff and Kingman

## TL;DR

This paper simplifies proofs of classical ergodic theorems by exploiting symmetry in Hopf's ratio ergodic theorem and extends ratio ergodic results to conservative transformations, generalizing Kingman's theorem.

## Contribution

It provides a simplified proof framework for Hopf and Birkhoff's ergodic theorems and extends ratio ergodic theorems to broader classes of transformations.

## Key findings

- Simplified proofs of Hopf and Birkhoff's ergodic theorems.
- A new ratio ergodic theorem for conservative transformations.
- Generalization of Kingman's ergodic theorem for subadditive sequences.

## Abstract

Hopf's ratio ergodic theorem has an inherent symmetry which we exploit to provide a simplification of standard proofs of Hopf's and Birkhoff's ergodic theorems. We also present a ratio ergodic theorem for conservative transformations on a $\sigma$-finite measure space, generalizing Kingman's ergodic theorem for subadditive sequences and generalizing previous results by Akcoglu and Sucheston.