The effect of 'diffuser' coils which perturb the magnetic surfaces in a Tokamak-as in an 'ergodic limiter'-is investigated by means of a simple model. This model leads to an area-preserving two parameter map with interesting invariance properties which describes the production of magnetic islands and of an ergodic layer. As the strength of the diffuser is increased the magnetic islands remain fixed by change their size in such a way that the overall pattern, and the boundary of the ergodic zone, move away from the diffuser. Analytic expressions are derived for this movement and for the location of the ergodic boundary.