Introduction

Since a few years integrable discrete-time systems have attracted considerable attention. They have appeared in various forms: partial difference equations (lattice equations), integrable mappings, and discrete Painlevé equations. They have intriguing new applications, for instance in discrete geometry, quantum theory of conformal models, and cellular automata.

In cf. also, integrable multidimensional mappings were considered as arising from finite-dimensional reductions of lattice equations. This reduction from what are partial difference analogues of the KdV equation naturally led to the establishment of the important integrability characteristics: Lax pairs, classical r-matrix structures including a Lagrangian formalism. The Liouville integrability of the resulting mappings, in the spirit of, was established in in the sense that the discrete-time flow is the iterate of a canonical transformation, preserving a suitable symplectic structure, possessing invariants which are in involution with respect to this symplectic form. According to, and in complete analogy with the continuous-time situation, it follows that the discrete-time flow can thus be linearized on a hypertorus which is the intersection of the level sets of the invariants. However, the explicit parametrization of the map represents in itself a separate problem which was not pursued to the end.

In this note we address this latter problem. We concentrate on the multidimensional rational mappings studied in, which form the multidimensional generalizations of the well-known McMillan map, and show that they can be parametrized (uniformized) by hyperelliptic abelian functions.