Invariant Measures for the n-Dimensional Border Collision Normal Form

Abstract

The border collision normal form is a continuous piecewise affine map of ℝn with applications in piecewise smooth bifurcation theory. We show that these maps have absolutely continuous invariant measures for an open set of parameter space and hence the attractors have Hausdorff (fractal) dimension n. If n = 2 the attractors have topological dimension two, i.e. they contain open sets, and if n > 2 then they have topological dimension n generically.