TAILIEUCHUNG - Đề tài " Stability of mixing and rapid mixing for hyperbolic flows "

We obtain general results on the stability of mixing and rapid mixing (superpolynomial decay of correlations) for hyperbolic flows. Amongst C r Axiom A flows, r ≥ 2, we show that there is a C 2 -open, C r -dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing. This is the first general result on the stability of rapid mixing (or even mixing) for Axiom A flows that holds in a C r , as opposed to H¨lder, topology. o | Annals of Mathematics Stability of mixing and rapid mixing for hyperbolic flows By Michael Field Ian Melbourne and Andrei T of ok Annals of Mathematics 166 2007 269 291 Stability of mixing and rapid mixing for hyperbolic flows By Michael Field Ian Melbourne and Andrei Torok Abstract We obtain general results on the stability of mixing and rapid mixing superpolynomial decay of correlations for hyperbolic flows. Amongst Cr Axiom A flows r 2 we show that there is a C2-open Cr-dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing. This is the first general result on the stability of rapid mixing or even mixing for Axiom A flows that holds in a Cr as opposed to Holder topology. 1. Introduction Let M be a compact connected differential manifold and let Tt be a C1 flow on M. A Tt-invariant set A is topologically mixing if for any nonempty open sets U V c A there exists T 0 such that Tt U n V 0 for all t T. The flow is stably mixing if all nearby flows in an appropriate topology are mixing. In this work we are interested in the Cr-stability of mixing and of the rate of mixing for Axiom A and Anosov flows. There is a quite extensive literature on mixing and rates of mixing for certain classes of Anosov flows. In particular Anosov 1 showed that geodesic flows for negatively curved compact Riemannian manifolds are always mixing. Anosov also proved the Anosov alternative a transitive volume-preserving Anosov flow is either mixing or the suspension of an Anosov diffeomorphism by a constant roof function. Plante 25 generalized the Anosov alternative to general equilibrium states and proved that codimension-one Anosov flows are mixing if and only if they are stably mixing for this class mixing is equivalent to the joint nonintegrability of the stable and unstable foliations which is a C 1-open condition . Anosov s results on geodesic flows were generalized to contact flows by Katok and Burns 19 . More recently Chernov 10 Dolgopyat Research supported in

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