Real saddle-node bifurcation from a complex viewpoint

M. Misiurewicz and R. A. Perez

Abstract

During a saddle-node bifurcation for real analytic interval maps, a pair of fixed points, attracting and repelling, collide and disappear. From the complex point of view, they do not disappear, but just become complex conjugate. The question is whether those new complex fixed points are attracting or repelling. We prove that this depends on the Schwarzian derivative S at the bifurcating fixed point. If S is positive, both fixed points are attracting, if it is negative, they are repelling.