This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows — they are either suspensions of Anosov diffeomorphisms or the stable and unstable distributions have equal dimensions. In particular, this proves that the examples in dimension strictly greater than 3 described by Franks and Williams in Anomalous Anosov Flows cannot be Anosov flows. In the second part, we give a new surgery type construction of Anosov flows, based on Franks and Williams 3-dimensional example, which yields non-transitive Anosov flows in all odd dimensions.
All Science Journal Classification (ASJC) codes