更高阶映射去自治化的困境
The trouble with deautonomising higher order maps
AI总结：
本文探讨了具有非受限奇点的更高阶映射去自治化的挑战，并提出了一种基于超离散映射的新方法来计算奇点乘数增长。
摘要：
将具有奇点限制性质的有理映射去自治化，即构造保持原映射奇点性质的非自治版本，已被证明在理解第二阶映射可积性背后的数学性质中至关重要。例如，去自治化过程直接促成了离散Painlevé方程的一般理论的发展，而且似乎很可能在任何未来更高维Painlevé方程的理论中也扮演关键角色。然而，一般来说，更高阶可积映射可能具有非受限奇点，因此重要的是要理解去自治化是否以及如何适用于此类映射。在本文中，我们探讨了不同去自治化场景下的若干精心构造的更高阶映射，这些映射既可积又不可积，并且具有非受限奇点，我们还挑战了一些关于可能在去自治化过程中起作用的奇点余维数的常见假设。在此过程中，我们还提出了一种新的计算出现在所谓的反受限奇点模式中的奇点乘数增长的程序，基于映射的超离散版本。
英文摘要：
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.
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