Detecting discrete integrability: the singularity approach

TL;DR

This paper explores how analyzing singularities in second order rational mappings helps detect integrability and compute dynamical degrees, emphasizing the practical role of singularity confinement.

Contribution

It provides a detailed classification of singularities and demonstrates how singularity analysis can explicitly determine the dynamical degree of mappings.

Findings

Singularity confinement serves as an effective integrability detector.

Explicit calculation of dynamical degrees using singularity analysis.

Classification of singularities in rational mappings.

Abstract

We describe the various types of singularities that can arise for second order rational mappings and we discuss the historical and present-day, practical, role the singularity confinement property plays as an integrability detector. In particular, we show how singularity analysis can be used to calculate explicitly the dynamical degree for such mappings.