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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.
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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.
We express discrete Painleve equations as discrete Hamiltonian systems. The discrete Hamiltonian systems here mean the canonical transformations defined by generating functions. Our construction relies on the classification of the discrete Painleve equations based on the surface-type. The discrete Hamiltonians we obtain are written in the logarithm and dilogarithm functions.
In this paper we study a certain recurrence relation, that can be used to generate ladder operators for the Laguerre Unitary ensemble, from the point of view of Sakais geometric theory of Painleve equations. On one hand, this gives us one more detailed example of the appearance of discrete Painleve equations in the theory of orthogonal polynomials. On the other hand, it serves as a good illustration of the effectiveness of a recently proposed procedure on how to reduce such recurrences to some canonical discrete Painleve equations.
The classical Painleve equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the complex domain. Where asymptotic descriptions are known, they are stated in the literature as valid for large connected domains, which include movable poles of families of solutions. However, asymptotic analysis necessarily assumes that the solutions are bounded and so these domains must be punctured at locations corresponding to movable poles, leading to asymptotic results that may not be uniformly valid. To overcome these issues, we recently carried out asymptotic analysis in Okamotos geometric space of initial values for the first and second Painleve equations. In this paper, we review this method and indicate how it may be extended to the discrete Painleve equations.
Four 4-dimensional Painleve-type equations are obtained by isomonodromic deformation of Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painleve system. Degenerating these four source equations, we systematically obtained other 4-dimensional Painleve-type equations. If we only consider Painleve-type equations whose associated linear equations are of unramified type, there are 22 types of 4-dimensional Painleve-type equations: 9 of them are partial differential equations, 13 of them are ordinary differential equations. Some well-known equations such as Noumi-Yamada systems are included in this list. They are written as Hamiltonian systems, and their Hamiltonians are neatly written using Hamiltonians of the classical Painleve equations.
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