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The entropy conjecture for diffeomorphisms (see for example this paper) asserts that for diffeomorphisms of manifolds, the log of the spectral radius of the actions of the diffeomorphism on the homology is a lower bound of the entropy of the diffeomorphism.

For flows, since they are naturally isotopic to the identity this concept makes no sense (except, for example, as noticed in the comments, when there is a global section).

However, we know that a flow $C^0$-close to an Anosov flow has positive entropy (this follows from the shadowing lemma), so I was wondering if there is a context where some kind of similar statement may hold.

Maybe the question is kind of vague, another way of asking would be: Is there a topological condition which guaranties that a flow has positive topological entropy? Preferably, I would like to have some condition which is in some sense global: For example, it should be $C^0$-open and not capture entropy created in euclidean balls (I would like to define this better, but I cannot find a way).

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This is really vague. Which "previous question" are you referring to? The only thing that comes to mind is that if you have a global section then you are in business, if not maybe it makes sense to take several sections and ask for some sort of markov property (the weaker the better). Some readily available references on entropy conjecture 1) 2) – Andrey Gogolev Sep 23 '10 at 20:25
The previous "question" would be if there is an analogue of the entropy conjecture for flows. Markov properties sound more local to me, essentially, I would prefer some global conditions (similar to the spectral radius of the action of homology, for flows with global sections it is clear how to define this, but what about other flows, for example for geodesic flows). – rpotrie Sep 23 '10 at 21:10
BTW, thanks for the first reference, I didn't know it. I will include it in the question as reference. – rpotrie Sep 24 '10 at 8:47
this restriction on comment length is no good – Andrey Gogolev Sep 24 '10 at 19:47
It seems like you would like to ignore flow on simple manifolds like spheres. Just a wild idea: take a cycle, flow it for some time. Then perturb the cycle (cut and reglue, but on a small set, not touching the cycle outside the small set). The result is new cycle. If there is some grouth then we say that we see some topological growth. Say, your homology is Z^2, you start with (0,1) and end up with (2,10). As it is, it's not going to work for sure (think of horocycle flow in negative curvature), but a some quantitative – Andrey Gogolev Sep 24 '10 at 20:01

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