Singularity confinement for maps with the Laurent property

The singularity confinement test is very useful for isolating integrable cases of discrete-time dynamical systems, but it does not provide a sufficient criterion for integrability. Quite recently a new property of the bilinear equations appearing in discrete soliton theory has been noticed: the iterates of such equations are Laurent polynomials in the initial data. A large class of non-integrable map-pings of the plane are presented which both possess this Laurent property and have confined singularities. There continues to be a great deal of interest in discrete-time dynamical systems that are integrable. There is a vast range of such systems, including symplectic maps and Bäcklund transformations for Hamiltonian systems in classical mechanics [1], mappings that preserve plane curves [2] which occur in statistical mechanics, discrete analogues of Painlevé transcendents [3], partial difference soliton equations appearing in numerical analysis and solvable quantum models [4], and equations arising in theories of discrete geometry and discrete analytic functions [5]. For some