Period-Adding Bifurcations in a One Parameter Family of Interval Maps

The dynamics and bifurcations of a one-parameter family of interval maps with a single jump discontinuity are studied. The central assumptions are that the maps are injective, increasing to the left of the discontinuity and decreasing to the right. It is shown that such maps have only periodic orbits of periods n; n +1; 2n, and 2n+2 with at least one of these being attracting. The value of n depends on the preimages of the discontinuity. Two types of bifurcations occur when an iterate of the discontinuity is mapped onto itself. In one type of bifurcation, a period n + 1 orbit emerges to coexist with a preexisting period n orbit. In the other type, the period n orbit disappears leaving a period n + 1 orbit. Generically, these two bifurcations must alternate, giving regions of coexistence and a global period-adding genealogy.