The redemption of singularity confinement

TL;DR

The paper introduces a new method using full deautonomisation to apply singularity confinement as a criterion for discrete integrability, enabling exact algebraic entropy computation and distinction between integrable and non-integrable mappings.

Contribution

It proposes a novel full deautonomisation approach that enhances singularity confinement as a discrete integrability test, applicable to complex mappings including the Hietarinta-Viallet example.

Findings

The method accurately computes algebraic entropy for various mappings.

It distinguishes integrable from non-integrable cases with confined singularities.

The approach is effective on well-known examples like the Hietarinta-Viallet mapping.

Abstract

We present a novel way to apply the singularity confinement property as a discrete integrability criterion. We shall use what we call a full deautonomisation approach, which consists in treating the free parameters in the mapping as functions of the independent variable, applied to a mapping complemented with terms that are absent in the original mapping but which do not change the singularity structure. We shall show, on a host of examples including the well-known mapping of Hietarinta-Viallet, that our approach offers a way to compute the algebraic entropy for these mappings exactly, thereby allowing one to distinguish between the integrable and non-integrable cases even when both have confined singularities.