On Decay of Correlations in Anosov Flows

Abstract

There is some disagreement about the meaning of the phrase 'chaotic flow.' However, there is no doubt that mixing Anosov flows provides an example of such systems. Anosov systems were introduced and extensively studied in his classical memoir ([A]). Among other things he proved the following fact known now as Anosov alternative for flows: Either every strong stable and strong unstable manifold is everywhere dense or the flow gt is a suspension over an Anosov diffeomorphism by a constant roof function. If the first alternative holds gt is mixing with respect to every Gibbs measure (see [PP2]). Now the natural question is: What is the estimated rate of mixing? This is certainly one of the simplest questions concerning correlation decay in continuous time systems. Nevertheless the only results obtained until recently dealt with the case when the system discussed had an additional algebraic structure. The easier case of Anosov diffeomorphisms can be treated by the methods of thermodynamic formalism of Sinai, Ruelle and Bowen ([B2]). Namely, one uses Markov partitions to construct an isomorphism between the diffeomorphism and a subshift of finite type and then proves that all such subshifts are exponentially mixing. This method would succeed also for flows if any suspension over a subshift of finite type had exponentially decaying correlations. However, the simplest example-suspensions with locally constant roof functions-never have such a property ([RI]). One can use the above observation to produce examples of Axiom A flows with arbitrary slow correlation decay. It became clear therefore that some additional geometric properties should be taken into account. In recent work Chernov ([Chl], [Ch2]) has employed a uniform nonintegrability condition to get a subexponential estimate for correlation functions for geodesic flows on surfaces of variable negative curvature. His method relies on the technique of Markov approximation developed in [Chl]. The aim of this paper is to combine geometric considerations of Chernov with the thermodynamic formalism approach. The later method seems to be more appropriate than Markov approximations since it gives simpler