Integrability of $n$-dimensional dynamical systems of type $E_7^{(1)}$ and $E_8^{(1)}$

Abstract

We propose an $n$-dimensional analogue of elliptic difference Painlevé equation. Some Weyl group acts on a family of rational varieties obtained by successive blow-ups at $m$ points in $\mpp^n(\mc)$, and in many cases they include the affine Weyl groups with symmetric Cartan matrices as subgroups. It is shown that the dynamical systems obtained by translations of these affine Weyl groups possess commuting flows and that their degrees grow quadratically. For the $E_7^{(1)}$ and $E_8^{(1)}$ cases, existence of preserved quantities is investigated. The elliptic difference case is also studied.

References in corpus (4)

• Discrete dynamical systems associated with root systems of indefinite type
• Algebraic Entropy of Birational Maps with Invariant Curves
• Painlevé equations and deformations of rational surfaces with rational double points
• Elliptic curves and birational representation of Weyl groups

Cited by in corpus (1)

• Tropical representation of Weyl groups associated with certain rational varieties