Singularity, complexity, and quasi--integrability of rational mappings

TL;DR

This paper analyzes the global properties and integrability conditions of rational mappings related to symmetries in integrable models, focusing on invariants, singularities, and dynamical complexity.

Contribution

It provides an algorithmic approach to analyze invariants and explores conditions for (quasi)--integrability linked to singularities in rational mappings.

Findings

Identification of invariants in rational mappings

Conditions for (quasi)--integrability related to singularities

Examples of complex dynamical behaviors

Abstract

We investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models, by exploiting their nature of birational transformations of projective spaces. We give an algorithmic analysis of the structure of invariants of such mappings. We discuss some characteristic conditions for their (quasi)--integrability, and in particular its links with their singularities (in the 2--plane). Finally, we describe some of their properties {\it qua\/} dynamical systems, making contact with Arnol'd's notion of complexity, and exemplify remarkable behaviours.