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

TL;DR

This paper demonstrates that automorphism groups of zero entropy infinite shifts have a natural homomorphism to the real numbers, derived from invariant measures, revealing structural properties of these dynamical systems.

Contribution

It introduces a novel 'drift' homomorphism from automorphism groups of zero entropy shifts to 5(5,5), based on cocycles and invariant measures, advancing understanding of their algebraic structure.

Findings

Existence of a 'drift' homomorphism for all zero entropy shifts

Homomorphism maps the shift to 1, linking dynamics to algebraic structure

Provides a new tool for analyzing automorphism groups in symbolic dynamics

Abstract

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.