Milnor attractors and topological attractors of a piecewise linear map

A very simple two-dimensional map is discussed. It is shown that for appropriate values of the parameters there is a two-dimensional subset of the plane on which the dynamics is transitive and periodic orbits are dense, but that this topological attractor contains a one-dimensional set which attracts almost all points (i.e, it is a Milnor attractor). This arises naturally as a precursor to a blowout bifurcation to on-off intermittency in this system, and confirms a conjecture due to Pikovsky and Grassberger.