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Anosov diffeomorphism (Definition)

If $ M$ is a compact smooth manifold, a diffeomorphism $ f\colon M\to M$ (or a flow $ \phi\colon\mathbb{R}\times M\to M$) such that the whole space $ M$ is an hyperbolic set for $ f$ (or $ \phi$) is called an Anosov diffeomorphism (or flow).

Anosov diffeomorphisms were introduced by D.V. Anosov, who proved that they are $ \mathcal{C}^1$-structurally stable.

Not every manifold admits an Anosov diffeomorphism; for example, there are no such diffeomorphisms on the sphere $ S^n$. The simplest examples of compact manifolds admiting them are the tori $ \mathbb{T}^n$: they admit the so called linear Anosov diffeomorphisms, which are isomorphisms of $ \mathbb{T}^n$ having no eigenvalue of modulus $ 1$. It was proved that any other Anosov diffeomorphism in $ \mathbb{T}^n$ is topologically conjugate to one of this kind.

It is not known which manifolds support Anosov diffeomorphisms. The only known examples of are nilmanifolds and infranilmanifolds, and it is conjectured that these are the only ones. Anosov flows are more abundant; for example, if $ M$ is a Riemannian manifold of negative sectional curvature, then its geodesic flow is an Anosov flow.

Another famous conjecture is that the nonwandering set of any Anosov diffeomorphism is the whole manifold $ M$. This is known to be true for linear Anosov diffeomorphisms (and hence for any Anosov diffeomorphism in a torus). For Anosov flows, there are examples where the nonwandering set is a proper subset of $ M$.



"Anosov diffeomorphism" is owned by Koro.
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Also defines:  Anosov flow
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Cross-references: proper subset, torus, nonwandering set, conjecture, geodesic, sectional curvature, negative, Riemannian manifold, support, topologically conjugate, modulus, eigenvalue, isomorphisms, sphere, stable, hyperbolic set, flow, diffeomorphism, smooth manifold, compact

This is version 6 of Anosov diffeomorphism, born on 2003-07-26, modified 2006-09-13.
Object id is 4511, canonical name is AnosovDiffeomorphism.
Accessed 5991 times total.

Classification:
AMS MSC: 37D20 (Dynamical systems and ergodic theory :: Dynamical systems with hyperbolic behavior :: Uniformly hyperbolic systems )
Pending Errata and Addenda
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Infranil manifolds & Riemann surfaces by Linas on 2005-09-09 00:47:04
I have in front of me a textbook example of an Anosov flow; its the spliting of the tangent manifold of the upper half plane (or of a Riemann surface of negative curvature) into three parts: the geodesic flow plus two horocycle flows, one expanding, the other contracting. I'm having some trouble identifying how this is an infranil manifold. I guess the group structure is rotations O(2) semi-direct translations; and te finite group must be the covering group. So I guess that means every tangent bundle to a Riemann surface is an infranil manifold ?? It'd be nice for this article to clarify. --linas
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