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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).

3 added 212 characters in body

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 .

However, (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 is still vaguebetter, but together with the previous question I think it makes some sense)cannot find a way).

2 added 141 characters in body

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.

However, for example, 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.

Sorry for

Maybe the question being is kind of vague, another way of asking would be: Is there a topological condition which guaranties that a flow has positive topological entropy? (this is still vague, but together with the previous question I prefer to leave think it this way so that different kind of answers can appearmakes some sense).

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