Abstract
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 codimension-one 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.
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