We show that a self orbit equivalence of a transitive Anosov flow on a 3-manifold which is homotopic to identity has to either preserve every orbit or the Anosov flow is R-covered and the orbit equivalence has to be of a specific type. This result shows that one can remove a relatively unnatural assumption in a result of Farrell and Gogolev [FG16] about the topological rigidity of bundles supporting a fiberwise Anosov flow when the fiber is 3-dimensional.

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Abstract. The main result of this article is that if a 3-manifold M supports an Anosov flow, then the number of conjugacy classes in the fundamental group of M grows exponentially fast with the length of the shortest orbit representative, hereby answering a question raised by Plante and Thurston in 1972. In fact we show that, when the flow is transitive, the exponential growth rate is exactly the topological entropy of the flow. We also show that taking only the shortest orbit representatives in each conjugacy classes still yields Bowen's version of the measure of maximal entropy. These results are achieved by obtaining counting results on the growth rate of the number of periodic orbits inside a free homotopy class. In the first part of the article, we also construct many examples of Anosov flows having some finite and some infinite free homotopy classes of periodic orbits, and we also give a characterization of algebraic Anosov flows as the only R-covered Anosov flows up to orbit equivalence that do not admit at least one infinite free homotopy class of periodic orbits.

Abstract. We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections or local coordinates. We give explicit representations and computations of spectral data for this operator in the case of Katok-Ziller metrics on the sphere and the torus.

We propose a generalization of the concept of discretized Anosov flows that covers a wide class of partially hyperbolic diffeomorphisms in 3manifolds, and that we call collapsed Anosov flows. They are related with Anosov flows via a self orbit equivalence of the flow. We show that all the examples from [BGHP] belong to this class, and that it is an open and closed class among partially hyperbolic diffeomorphisms. We provide some equivalent definitions which may be more amenable to analysis and are useful in different situations. Conversely we describe the class of partially hyperbolic diffeomorphisms that are collapsed Anosov flows associated with certain types of Anosov flows.