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