Algebraic Entropy for lattice equations

TL;DR

This paper introduces algebraic entropy as a measure of complexity for lattice equations, linking zero entropy to integrability and suggesting it takes special algebraic values.

Contribution

It defines algebraic entropy for lattice equations and proposes it as a tool for detecting integrability and understanding their complexity.

Findings

Algebraic entropy is a canonical measure of lattice equation complexity.

Vanishing entropy indicates integrability.

Entropy values are conjectured to be algebraic integers.

Abstract

We give the basic definition of algebraic entropy for lattice equations. The entropy is a canonical measure of the complexity of the dynamics they define. Its vanishing is a signal of integrability, and can be used as a powerful integrability detector. It is also conjectured to take remarkable values (algebraic integers).