Abstract

If the iterates of a rational map are Laurent polynomials in the initial data, then the map is said to have the Laurent property. This Laurent phenomenon is a feature of bilinear equations for tau-functions that appear in the theory of discrete integrable systems, but it has only recently been understood as a by-product of Fomin and Zelevinsky˜s theory of cluster algebras. Maps with the Laurent property are also of interest to number theorists, as they lead to integer sequences and hence to the solutions of Diophantine equations. We present a large class of non-integrable measure-preserving maps of the plane that have the Laurent property, and show the connection with the singularity confinement property introduced by Grammaticos, Ramani and Papageorgiou. We also explain how to solve various associated Diophantine problems.