A systematic method for constructing discrete Painlevé equations in the degeneration cascade of the E${}_8$ group

Abstract

We present a systematic and quite elementary method for constructing discrete Painlevé equations in the degeneration cascade for E${}_8^{(1)}$. Starting from the invariant for the autonomous limit of the E${}_8^{(1)}$ equation one wishes to study, the method relies on choosing simple homographies that will cast this invariant into certain judiciously chosen canonical forms. These new invariants lead to mappings the deautonomisations of which allow us to build up the entire degeneration cascade of the original mapping. We explain the method on three examples, two symmetric mappings and an asymmetric one, and we discuss the link between our results and the known geometric structure of these mappings.

22 pages, 5 figures

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Cited by in corpus (4)

• Space of initial conditions and geometry of two 4-dimensional discrete Painlevé equations
• Restoring discrete Painlevé equations from an E${}_8^{(1)}$-associated one
• Discrete Painlevé equations from singularity patterns: the asymmetric trihomographic case
• Detecting discrete integrability: the singularity approach