Contact Anosov flows on hyperbolic 3–manifolds

Abstract

Geodesic flows of Riemannian or Finsler manifolds have been the only known contact Anosov flows. We show that even in dimension 3 the world of contact Anosov flows is vastly larger via a surgery construction near an E-transverse Legendrian link that encompasses both the Handel-Thurston and Goodman surgeries and that produces flows not topologically orbit equivalent to any algebraic flow. This includes examples on many hyperbolic 3-manifolds, any of which have remarkable dynamical and geometric properties. To the latter end we include a proof of a folklore theorem from 3-manifold topology: In the unit tangent bundle of a hyperbolic surface, the complement of a knot that projects to a filling geodesic is a hyperbolic 3-manifold.

Open sets of exponentially mixing Anosov flows

“…Ratner [30] showed that the boundary of the Markov partition has Lebesgue measure zero but this is not quite sufficient for our purposes. Fortunately, as shown by Horita & Viana [26,Proposition 3.5] we also have estimates for the box-counting dimension 22 of the boundary. Section A.1 is devoted to reviewing this topic and the information on the dimension of the boundary is a key point in constructing the partition of unity of Lemma 3.10.…”

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

“…The non-transversely orientable case is obtained as a corollary of the main theorem. Proof of Corollary 2 assuming Theorem 1: In [14] Foulon and Hasselblatt showed that there exist infinitely many non-diffeomorphic hyperbolic 3-manifolds that admit contact forms whose Reeb flows are transversely orientable Anosov flows. This combined with Theorem 2 implies the corollary.…”

Knot theory of ℝ-covered Anosov flows: homotopy versus isotopy of closed orbits

“…If the manifold is also atoroidal, then these flows satisfy the homotopic properties of closed orbits described above. Note that the contact Anosov flows constructed in [13] are of this type.…”