Homomorphisms to $\mathbb{R}$ of automorphism groups of zero entropy shifts

Omer Tamuz

Published 2021-12-14, updated 2022-02-18Version 3

We show that the automorphism group of every zero entropy infinite shift admits a "drift" homomorphism to $(\mathbb{R},+)$ that maps the shift map to 1. This homomorphism arises as the expectation, under an invariant measure, of a cocycle defined on a space of asymptotic pairs.