A systematic method for constructing discrete Painlev\'e equations in the degeneration cascade of the E$_8$ group

TL;DR

This paper introduces a straightforward method to systematically construct discrete Painlevé equations within the E8 degeneration cascade, utilizing invariants and homographies to generate mappings and explore their geometric structures.

Contribution

The paper presents an elementary, systematic approach to derive discrete Painlevé equations in the E8 degeneration cascade using invariants and homographies, expanding the toolkit for studying these equations.

Findings

Successfully applied the method to three examples, including symmetric and asymmetric mappings.

Established links between the constructed mappings and their geometric structures.

Demonstrated the method's effectiveness in generating the entire degeneration cascade.

Abstract

We present a systematic and quite elementary method for constructing discrete Painlev\'e 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.