CkSmoothness of Invariant Curves in a Global Saddle-Node Bifurcation

Abstract

The birth ofCk-smooth invariant curves from a saddle-node bifurcation in a family ofCkdiffeomorphisms on a Banach manifold (possibly infinite dimensional) is constructed in the case that the fixed point is a stable node along hyperbolic directions, and has a smooth noncritical curve of homoclinic orbits. This ensures that the map restricted to the resulting curve is equivalent to aCkmap of the circle. In particular, for aC2family of diffeomorphisms the resulting curve isC2, and the “Denjoy example” cannot occur. Included is a new smoothness result for the foliation transversal to the center subspace, for the finite and infinite dimensional cases. Specifically,Ck-smoothness with respect to all variables of invariant foliations of the center-stable and center-unstable manifolds of a partially hyperbolic fixed point is proved in all cases.