Singularity confinement as an integrability criterion

TL;DR

This paper introduces a rigorous method to determine the integrability of birational three point mappings with singularity confinement, using their singularity patterns, and explains how to derive their dynamical degree from these patterns.

Contribution

It provides a novel, structure-based criterion for integrability and a way to compute the dynamical degree from singularity patterns.

Findings

Method effectively distinguishes integrable from non-integrable mappings.

Dynamical degree can be deduced directly from singularity patterns.

Provides a rigorous framework for analyzing singularity confinement.

Abstract

In this paper we present a rigorous method for deciding whether a birational three point mapping that has the singularity confinement property is integrable or not, based only on the structure of its (confined) singularity patterns. We also explain how the exact value of the dynamical degree for such a mapping may be deduced from the singularity patterns.