A \(C^1\)-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks of sources
type: Item
instance of: scholarly article
title: A \(C^1\)-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks of sources
zbMATH Open document ID: 1049.37011
author: C. Bonatti
author: L. J. Díaz
author: Enrique R. Pujals
published in: Annals of Mathematics. Second Series
publication date: 2004-03-07T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.annm/1069786250
review text: \textit{R. Mañé} [Ann. Math. (2) 116, 503--540 (1982; Zbl 0511.58029)] proved that in the \(C^1\) topology, there is a dichotomy in possible dynamics for surface diffeomorphisms, for a residual set of diffeomorphisms. A diffeomorphism from the residual set either has infinitely many sinks or sources, or is Axiom A. An extension of this result to more dimensions is not straightforward, because of examples, for instance on three- and four-dimensional tori, of \(C^1\) robustly transitive but nonhyperbolic diffeomorphisms. The authors prove a generalization of Mañé's result replacing the notion of hyperbolicity by the existence of dominated splittings. A dominated splitting on an invariant set \(K\) for a diffeomorphism \(f\) on a manifold \(M\) is an \(f_*\) invariant splitting \(TM = E \oplus F\) with fibers of constant dimension and \[ \| f^n_* (x)| _E \| \| f^{-n}_* (f^n(x))| _F \| < 1/2 \] for \(n\) large enough. The splitting is automatically continuous. A dominated splitting on a maximal invariant set is a \(C^1\) robust property. Consider a hyperbolic saddle \(P\) for \(f\). Its homoclinic class \(H(P,f)\) is the closure of the set of transverse intersections of its stable and unstable manifolds. It is shown that for \(f\) from a residual subset of diffeomorphisms on \(M\), in the \(C^1\) topology, the following holds. Let \(P\) be a hyperbolic saddle of \(f\). Then either \(H(P,f)\) admits a dominated splitting or \(H(P,f)\) is in the closure of infinitely many sinks and sources of \(f\). As a consequence, any \(C^1\) robust transitive diffeomorphism admits a dominated splitting. A long term objective is to get a spectral decomposition theorem of the nonwandering set for nonhyperbolic diffeomorphisms away from the Newhouse phenomenon of infinitely many sinks or sources. One could for instance try to establish that the nonwandering set of diffeomorphisms away from the Newhouse phenomenon is the union of finitely many disjoint compact invariant sets admitting a d…
reviewed by: Ale Jan Homburg
Mathematics Subject Classification ID: 37C20
Mathematics Subject Classification ID: 37D30
zbMATH DE Number: 2052016
zbMATH Keywords: dichotomy
zbMATH Keywords: dominated splittings
zbMATH Keywords: \(C^1\) robust property
zbMATH Keywords: spectral decomposition
zbMATH Keywords: nonwandering set
zbMATH Keywords: nonhyperbolic diffeomorphisms
zbMATH Keywords: Newhouse phenomenon
zbMATH Keywords: volume preserving diffeomorphisms
MaRDI profile type: Publication
OpenAlex ID: W2106627783
DOI: 10.4007/ANNALS.2003.158.355
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description: scientific article; zbMATH DE number 2052016