The real analytic Feigenbaum-Coullet-Tresser attractor in the disk

Details The real analytic Feigenbaum-Coullet-Tresser attractor in the disk The real analytic Feigenbaum-Coullet-Tresser attractor in the disk

2008-05-08

arxiv.org/abs/0805.1152v1

Dynamical Systems Vol. 25, No. 4, December 2010, 519-529

Mathematics

Dynamical Systems

Scientific paper

10.1080/14689361003785727

We consider a real analytic diffeomorphism $\psi_0$ on a n-dimensional disk D, n >= 2, exhibiting a Feigenbaum-Coullet-Tresser (F.C.T.) attractor, being far, in the standard topology of the real analytic diffeomorphism space C(D), from the standard F.C.T. map $\phi_0$ fixed by the double renormalization. We prove that $\psi_0$ persists along a codimension-one manifold M \subset C(D), and that it is the bifurcating map along any one-parameter family in $C(D)$ transversal to M, from diffeomorphisms attracted to sinks, to those which exhibit chaos. The main tool in the proofs is a theorem of Functional Analysis, which we state and prove in this paper, characterizing the existence of codimension one submanifolds in any abstract functional Banach space.

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