CkSmoothness of Invariant Curves in a Global Saddle-Node Bifurcation
✍ Scribed by Todd Young
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 574 KB
- Volume
- 126
- Category
- Article
- ISSN
- 0022-0396
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✦ Synopsis
The birth of C k -smooth invariant curves from a saddle-node bifurcation in a family of C k diffeomorphisms 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 a C k map of the circle. In particular, for a C 2 family of diffeomorphisms the resulting curve is C 2 , 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, C k -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.
1996 Academic Press, Inc. and [AAIS]). We prove in this article C k -smoothness of the invariant curve if the original family is C k -smooth under general assumptions.
For this bifurcation the case k=2 (or k=1 with bounded variation of the derivative) is of special importance due to the Denjoy Theory [De, Ni] for diffeomorphisms of the circle, S 1 . If it is known that the resulting article no. 0044