Garden of Eden and Specification

TL;DR

This paper proves a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, linking pre-injectivity and surjectivity of continuous equivariant maps.

Contribution

It establishes a new Garden of Eden theorem for a broad class of algebraic actions, extending previous results to include expansive principal algebraic actions and actions of Z^d with CPE.

Findings

Pre-injective maps are surjective for the considered class of actions.

The theorem applies to all expansive principal algebraic actions of amenable groups.

It generalizes the Garden of Eden theorem to new algebraic dynamical systems.

Abstract

We establish a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, i.e. for any continuous equivariant map T from the underlying space to itself, T is pre-injective if and only if it is surjective. In particular, this applies to all expansive principal algebraic actions of amenable groups and expansive algebraic actions of Z^d with CPE.