Talk:Anosov diffeomorphism
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[edit] SL(2,R)
The standard geodesic and horocycle flows on SL(2,R) are Anosov flows. This article then implies that SL(2,R) is an infranil manifold ?? linas 02:48, 6 September 2005 (UTC)
- OK, well, I just found the definition of infranil manifold, so I guess the answer is yes. Err, sl(2,R) is nilpotent, I guess? We need a list of nilpotent Lie algebras. linas 04:12, 6 September 2005 (UTC)
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- No, not nilpotent (Engel's theorem fails or something). Charles Matthews 16:11, 6 September 2005 (UTC)
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- Well, I didn't think it was. As the book I have in front of me explicitly develops the Anosov flows for SL(2,R), I'll need to re-read this article, and the definition of infranil, as there seems to be a contradiction/error somewhere. linas 00:33, 7 September 2005 (UTC)
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- OK, after reading the definition more carefully, I'm starting to guess that the tangent bundle to every Riemann surface of 0 or negative curvature is an infranil manifold. (The geodesic flow on this manifold is the part that's neither contracting nor expanding; there are only two other flows, one contracts, one expands).
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- Is it possible that tangent manifolds in general are infranil? i.e. the nilpotent group is then the group of motions on an n-dimensional tangent bundle, which would be the rotations O(n) and semi-direct product of geodesic flow. Maybe I have the definition of infranil upside down, still. linas 05:10, 9 September 2005 (UTC)
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[edit] Thinking aloud
Just wanted to set down some thoughts/conjectures. Following the example of SL(2,R) in the article, if g is a Lie algebra, with a one-dimensional subspace, call it j, and two other subspaces, called x and y, with algebra
- [j,x]=+x and [j,y]= -y
where + and - here means that the structure constants are all positive, and all negative, then we can use j to define a 1-parameter flow on the manifold of the Lie group G. It seems to me that this would meet the criteria for being an Anosov flow. Furthermore, given just about any lattice Γ in G could be used to define a manifold G\Γ which would also have that flow. Right? Or am I missing something? What other Lie algebras, besides SL(2,R), that have structure constants like this? Can this be extended to more general algebras? How about infinite-dimensional algebras, e.g. C-star algebras? Hmm ... quantum mechanics ... linas 05:16, 11 September 2005 (UTC)
User:R.e.b reminded me that:
- The heisenberg algebra and the virasoro algebra (without the center) and the elements of degree at least -1 of the latter algebra are three more examples. If you dont mind y being 0 you can take x to be any nilpotent algebra and j a suitable outer derivation. There are also a few Lie superalgebras with this property.
[edit] Flows vs. diffeomorphisms
Pardon me if I'm missing something, but I'm wondering why this page concentrates so much on Anosov flows. As I understand it, Anosov diffeomorphisms are significantly different to Anosov flows: Anosov diffeomorphisms act nontrivially on the homotopy and homology groups of the manifold, whereas Anosov flows, being homotopies, do not; and the time-one map of an Anosov flow is never an Anosov diffeomorphism. As such, I'm not sure how relevant to this article Anosov flows actually are. Shouldn't they be in a separate article? --Invisible Capybara 17:30, 29 July 2006 (UTC)
