Counting periodic orbits of Anosov flows in free homotopy classes

Abstract

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.

A note on self orbit equivalences of Anosov flows and bundles with fiberwise Anosov flows

“…We want to show that h( α) = α. If F H( α) is not a string of lozenges, i.e., there exists two corners on non-separated leaves (see [BF15b, Section 2]), then we already showed that α is fixed. So we may assume that F H( α) is a string of lozenges.…”

“…In order to keep this note short, we will refer to Section 2 of [BF15b] for all the necessary background information on Anosov flows in dimension 3. Notice that there is one important difference of convention: in [BF15b], free homotopy between orbits refers to a free homotopy between the non-oriented curves represented by the orbits, whereas in this present note, we do not forget the orientation.…”

“…The reason one would want to remove the assumption b) of Farrell-Gogolev's result is that it seems to be a quite unnatural condition, as it does not appear to be verified very often. Indeed, in dimension 3, Barthelmé and Fenley [BF15b] show that lots of (or even "most") examples have at least some free homotopy class with infinitely many distinct orbits. In fact a (conjecturally) complete list of types of 3-dimensional Anosov flows that satisfy assumption b) is as follows:…”

Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds

“…This theorem establishes the positive of topological entropy for all Reeb flows on an Anosov contact 3-manifold (M, ξ) provided that (M, ξ) admits a transversely orientable Anosov Reeb flow. The main tools in the proof of the theorem are the results of Fenley [13] and, most importantly, the recent work of Barthelmé and Fenley [4]. The non-transversely orientable case is obtained as a corollary of the main theorem.…”

“…Crucial for the proof of Theorem 2 is the recent work of Barthelmé and Fenley [4]. They proved that the number of free homotopy classes in a compact 3-manifold containing periodic orbits with action ≤ T of an Anosov flow grow exponentially with T .…”