The real analytic Feigenbaum-Coullet-Tresser attractor in the disc.

We consider a real analytic diffeomorphism ψ0 on an n-dimensional disc D, n ≥ 2, exhibiting a Feigenbaum-Coullet-Tresser (FCT) attractor. We assume that in the C ω(D) topology it is far from the standard FCT map ϕ0 fixed by the double renormalization. We prove that ψ0 persists along a codimensionone manifold M ⊂ C ω(D), and that it is the bifurcating map along any one-parameter family in C ω(D) transversal to M, from diffeomorphisms exhibiting sinks to those which exhibit chaos, filling a gap in the usually accepted proof of this assertion. The main tool in the proofs is a theorem of functional analysis, which we state and prove in this article, characterizing the existence of codimension-one submanifolds in any abstract functional Banach space.