The trouble with deautonomising higher order maps

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 Painlevé equations, and it seems highly likely it will play a crucial role in any future theory of higher dimensional Painlevé 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.

💡 Research Summary

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The paper investigates the deautonomisation (or “full‑deautonomisation”) of birational maps that possess non‑confined singularities, focusing on higher‑order (order three and above) mappings. While the deautonomisation procedure based on singularity confinement has been highly successful for second‑order maps—leading to the systematic derivation of discrete Painlevé equations—the situation is far less clear for higher‑order maps where singularities often fail to confine. The authors aim to understand whether, and how, deautonomisation can be applied when only a few confined singularities coexist with a plethora of non‑confined or anticonfined ones.

The study proceeds by constructing a series of explicit higher‑order examples. The basic building block is the integrable QRT map
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