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