Generalization of a Theorem of Malta and Palis
Young, Todd R.
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We study the saddle-node bifurcation of a partially hyperbolic fixed point in a Lipschitz family of C^k diffeomorphisms on an n-dimensional manifold, in the case that the stable set and unstable set of the fixed point intersect the stable and unstable manifolds of other invariant sets in a 'critical' manner. Sufficient conditions are found which guarantee realization of only codimension one bifurcations in a family. A diffeomorphism for which the conditions are not satisfied is shown to be in the closure of the set of codimension 2 bifurcation surfaces. These results are a generalization of a one-dimensional result of Malta and Palis.