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Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. (English) Zbl 0725.58034
The main geometric objects associated with an Anosov dynamical system on a compact manifold are the invariant stable and unstable foliations. Although each stable and unstable manifold is as smooth as the system itself, it is believed that the foliations that they form have only a moderate degree of regularity for most systems. In the paper under review, the authors analyze the exact degree of regularity for the weak-stable and weak-unstable foliations for a volume-preserving Anosov system of codimension one. The main results relate the regularity of these foliations to cohomology classes associated to the system: the Anosov class, a new invariant of the flow which is introduced in this paper, and the Godbillon-Vey class of the weak-stable foliation, which the authors show that it is a well-defined invariant of the system.
The general outline of this paper is based on the authors’ study of the local obstacles discovered by Anosov for the two-torus [D. V. Anosov, Tr. Mat. Inst. Steklova 90 (1967; Zbl 0163.43604)]. The authors show that the Anosov class has a remarkable application: when this cohomology “obstacle to regularity” vanishes, the foliations must be infinitely smooth (this is the rigidity phenomenon mentioned in the title). This implies that the system is algebraic by results of A. Avez [Topol. Dynamics, Int. Symp. Colorado State Univ. 1967, 17–51 (1968; Zbl 0203.26101)] or E. Ghys [Ann. Sci. Éc. Norm. Supér. (4) 20, 251–270 (1987; Zbl 0663.58025)]. The Godbillon-Vey invariant of the flow has two applications. Firstly, the authors show that there are continuous families of topologically conjugate, codimension-one foliations for which the Godbillon-Vey invariants vary continuously (and are not constant!). Secondly, the Godbillon-Vey invariant of the flow characterizes the geodesic flows for metrics of negative curvature as the flows with maximal value for this invariant, among the geodesic flows for metrics of negative curvature on closed surfaces. The authors use this observation to give a new proof that the harmonic measure at infinity for metrics of variable negative curvature on surfaces is totally singular.
Both of the cohomology invariants introduced in the paper are parameters on the space of low-dimensional, volume-preserving Anosov systems, for which the O-set is parametrized by Teichmüller spaces. The authors conjecture that for other values of these cohomology invariants, the system is determined up to smooth equivalence by a finite set of auxiliary, Teichmüller-like parameters.
Reviewer: D. Savin (Iaşi)

MSC:
37D99 Dynamical systems with hyperbolic behavior
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
37C10 Dynamics induced by flows and semiflows
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