Anosov Flows

Differentiability, rigidity and Godbillon-Vey classes for Anosov flows

“…The 3-form τ ∧dτ is continuous and flow-invariant. If it is identically zero, then by Plante (Theorem 3.1, [57]; see also Ghys [19]) there exists a compact smooth section for the flow which must be a 2-torus. Obviously, the section can be chosen to be C ∞ , from which the first case of (2.3.2) follows.…”

“…When the flow is topologically transitive, there is a unique continuous 1-form, up to a multiplicative constant, which vanishes on the subbundle E − ⊕E + , so that in the condition (5) we need only require that τ (ξ) = 1 at one point of M. In general, the form τ is only Hölder continuous (see Theorem 4.1 of [57]). If the form τ happens to be C 1 , then it carries considerable additional information.…”

“…However, this conclusion is not known if only the weak-stable and weak-unstable foliations are given to be C ∞ . The examples of Plante [57] show that an algebraic Anosov flow can be perturbed by a time change, so that it has C ∞ weak-stable and weak-unstable foliations, but the strong-stable and strong-unstable foliations are not even differentiable.…”

“…The strong foliations are then as regular as the weak foliations. (A well-known example of Plante [57] shows that without this requirement, the strong foliations need not even be C 1 .) For example, such a form always exists for geodesic flows of compact Riemannian surfaces of negative curvature.…”

Hyperbolic Dynamical Systems

“…Indeed, we have the Anosov alternative: Transitive Anosov flows are either mixing or a suspension [Pl1] (in the volume-preserving case this is due to Anosov [A1]). To be more precise, if there is a nondense strong leaf or if y x { y y is integrable then the flow is a suspension.…”