There has been much interest in recent years in the properties of Piecewise Isometries, which are simple, discontinuous maps of the plane. Several examples from the field of engineering have been found where such maps provide a description of the dynamics – for instance, mixing of granular materials, and electronic circuits such as the sigma-delta modulator, the buck converter and digital filters (in particular, the behaviour of their overflow oscillations). In the electronic examples,  modelling via a piecewise isometry is unrealistic, since dissipation is explicitly removed: making the model more realistic by including dissipation leads instead to a piecewise contraction. These were proved in 2008 to be asymptotically periodic, raising questions about what else can be said about the dynamics of iterated piecewise contractions. Of particular interest are questions related to co-existing solutions and whether contraction implies bounds on the periodicity of the solutions, as well as studying the geometry of subsets of the parameter plane where such solutions exist.