Space of initial conditions and geometry of two 4-dimensional discrete Painlevé equations

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 Néron-Severi bilattices. It turns out that the deautonomised version of one of the mappings is a Bäcklund transformation of a direct product of the fourth Painlevé 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 $A_5^{(1)}$ Painlevé equation.

28 pages

References in corpus (1)

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

Cited by in corpus (5)

• On the singularity structure of the discrete KdV equation
• An exercise in experimental mathematics: calculation of the algebraic entropy of a map
• Singularities and growth of higher order discrete equations
• Super-QRT and 4D-mappings reduced from the lattice super-KdV equation
• Singularity confinement and proliferation of tau functions for a general differential-difference Sawada-Kotera equation