Partially Hyperbolic Sets from a Co-Dimension One Bifurcation

Abstract

We study the saddle-node bifurcation of a partially hyperbolic fixed point in a Lipschitz family of C^k diffeomorphisms on a Banach manifold (possibly infinite dimensional) in the case that the fixed point is a saddle along hyperbolic directions and has multiple curves of homoclinic orbits. We show that this bifurcation results in an invariant set which consists of a countable collection of closed invariant curves and an uncountable collection of nonclosed invariant curves which are the topological limits of the closed curves. In addition, it is shown that these curves are C^k-smooth and that this invariant set is uniformly partially hyperbolic.