Space of initial conditions and geometry of two 4-dimensional discrete Painlev\'e equations

TL;DR

This paper provides a geometric analysis of two 4D discrete Painlevé equations, revealing their symmetries, invariants, and growth rates through rational variety transformations, and links them to known Painlevé equations.

Contribution

It introduces a geometric framework for understanding two 4D discrete Painlevé equations, connecting them to affine Weyl group symmetries and known Painlevé types.

Findings

One mapping relates to a Bäcklund transformation of a product of Painlevé IV equations.

The other mapping corresponds to Noumi-Yamada's A_5^{(1)} Painlevé equation.

Degree growth rates and invariants are computed from Néron-Severi lattices.

Abstract

A geometric study of two 4-dimensional mappings is given. By the resolution of indeterminacy they are lifted to pseudo-automorphisms of rational varieties obtained from $({\mathbb P}^1)^4$ by blowing-up along sixteen 2-dimensional subvarieties. The symmetry groups, the invariants and the degree growth rates are computed from the linearisation on the corresponding N\'eron-Severi bilattices. It turns out that the deautonomised version of one of the mappings is a B\"acklund transformation of a direct product of the fourth Painlev\'e equation which has $A_2^{(1)}+A_2^{(1)}$ type affine Weyl group symmetry, while that of the other mapping is of Noumi-Yamada's $A_5^{(1)}$ Painlev\'e equation.