The trouble with deautonomising higher order maps

TL;DR

This paper investigates the process of deautonomisation in higher order maps with non-confined singularities, challenging existing assumptions and proposing new methods to analyze singularity growth, which is crucial for understanding integrability.

Contribution

It explores deautonomisation scenarios for higher order mappings with non-confined singularities and introduces a novel ultradiscrete method to analyze singularity growth.

Findings

Deautonomisation can be applied to higher order maps with non-confined singularities.

Common assumptions about singularity loci co-dimensionality may not hold universally.

A new ultradiscrete approach effectively measures growth in anticonfined singularity patterns.

Abstract

The deautonomisation of birational maps that have the singularity confinement property, i.e. the construction of nonautonomous versions of such maps that preserve the singularity properties of the original, has proven crucial in our understanding of the mathematical properties behind the integrability of second order maps. For example, the deautonomisation procedure led directly to the development of a general theory of discrete Painlev\'e equations, and it seems highly likely it will play a crucial role in any future theory of higher dimensional Painlev\'e equations as well. Generally speaking however, higher order integrable mappings may have non-confined singularities and it is important to understand if, and how, deautonomisation should work for such mappings. In this paper we explore different deautonomisation scenarios on a series of carefully constructed higher order mappings, integrable as well as non-integrable, that possess non-confined singularities and we challenge some common assumptions regarding the co-dimensionality of the singular loci that might play a role in the deautonomisation process. Along the way we also propose a novel procedure to calculate the growth of the multiplicities of singularities that appear in so-called anticonfined singularity patterns, based on an ultradiscrete version of the mapping.