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# Anosov diffeomorphism

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 $M$.

## Mathematics Subject Classification

37D20*no label found*

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## Comments

## Infranil manifolds & Riemann surfaces

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

## Re: Infranil manifolds & Riemann surfaces

The problem is that you're looking at an Anosov *flow*. The conjecture is that the only manifolds supporting Anosov *diffeomorphisms* are nilmanifolds or infranilmanifolds. Anosov flows are much more common; in fact the example you mention is a particular case of the more general fact that in any riemannian manifold with negative sectional curvature, the geodesic flow is in fact an Anosov flow.

I will add a comment on that.