Abstract
We show that any topologically transitive codimension-one Anosov flow on a closed manifold is topologically equivalent to a smooth Anosov flow that preserves a smooth volume. By a classical theorem due to Verjovsky, any higher-dimensional codimension-one Anosov flow is topologically transitive. Recently, Simić showed that any higher-dimensional codimension-one Anosov flow that preserves a smooth volume is topologically equivalent to the suspension of an Anosov diffeomorphism. Therefore, our result gives a complete classification of codimension-one Anosov flows up to topological equivalence in higher dimensions.
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An erratum to this article can be found at http://dx.doi.org/10.1007/s00222-009-0211-9
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Asaoka, M. On invariant volumes of codimension-one Anosov flows and the Verjovsky conjecture. Invent. math. 174, 435–462 (2008). https://doi.org/10.1007/s00222-008-0151-9
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DOI: https://doi.org/10.1007/s00222-008-0151-9