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

TL;DR

This paper proves the persistence of a Feigenbaum-Coullet-Tresser attractor in real analytic diffeomorphisms on disks, showing its stability along a codimension-one manifold and its bifurcation to chaos.

Contribution

It demonstrates the existence and persistence of a Feigenbaum-Coullet-Tresser attractor in higher-dimensional disks and characterizes its bifurcation behavior in the space of real analytic diffeomorphisms.

Findings

Feigenbaum-Coullet-Tresser attractor persists along a codimension-one manifold.

Bifurcation from sinks to chaos occurs along transversal families.

Main tool is a functional analysis theorem on codimension-one submanifolds.

Abstract

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.