Renormalization of the Two-Dimensional Border-Collision Normal Form

All nondegenerate, continuous, piecewise-linear maps on [Formula: see text] with two pieces are equivalent to a member of four-parameter family known as the two-dimensional border-collision normal form. This paper shows how powerful technique renormalization can be applied this and reveals previously undescribed bifurcation structure in succinct way. We partition parameter region where is exhibit chaos robustly into infinitely many subregions an explicit then show chaotic attractor has different numbers connected components subregions. The results rely careful analysis global dynamics operator. challenging because operator essentially quadratic map text].