An algebraically stable variety for a four-dimensional dynamical system reduced from the lattice super-KdV equation

In a prior paper the authors obtained a four-dimensional discrete integrable dynamical system by the traveling wave reduction from the lattice super-KdV equation in a case of finitely generated Grassmann algebra. The system is a coupling of a Quispel-Roberts-Thompson map and a linear map but does not satisfy the singularity confinement criterion. It was conjectured that the dynamical degree of this system grows quadratically. In this paper, constructing a rational variety where the system is lifted to an algebraically stable map and using the action of the map on the Picard lattice, we prove this conjecture. We also show that invariants can be found through the same technique.

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

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 Neron-Severi bilattices. It turns out that the deautonomised version of one of the mappings is a Backlund transformation of a direct product of the fourth Painleve equation which has $A_2^{(1)}+A_2^{(1)}$ type affine Weyl group symmetry, while that of the other mapping is of Noumi-Yamadas $A_5^{(1)}$ Painleve equation.

Fiber-dependent deautonomization of integrable 2D mappings and discrete Painleve equations

It is well known that two-dimensional mappings preserving a rational elliptic fibration, like the Quispel-Roberts-Thompson mappings, can be deautonomized to discrete Painleve equations. However, the dependence of this procedure on the choice of a particular elliptic fiber has not been sufficiently investigated. In this paper we establish a way of performing the deautonomization for a pair of an autonomous mapping and a fiber. %By choosing a particular Starting from a single autonomous mapping but varying the type of a chosen fiber, we obtain different types of discrete Painleve equations using this deautonomization procedure. We also introduce a technique for reconstructing a mapping from the knowledge of its induced action on the Picard group and some additional geometric data. This technique allows us to obtain factorized expressions of discrete Painleve equations, including the elliptic case. Further, by imposing certain restrictions on such non-autonomous mappings we obtain new and simple elliptic difference Painleve equations, including examples whose symmetry groups do not appear explicitly in Sakais classification.

A note on minimization of rational surfaces obtained from birational dynamical systems

In many cases rational surfaces obtained by desingularization of birational dynamical systems are not relatively minimal. We propose a method to obtain coordinates of relatively minimal rational surfaces by using blowing down structure. We apply this method to the study of various integrable or linearizable mappings, including discre