On decay of correlations in Anosov flows

@article{Dolgopyat1998OnDO,
  title={On decay of correlations in Anosov flows},
  author={Dmitry Dolgopyat},
  journal={Annals of Mathematics},
  year={1998},
  volume={147},
  pages={357-390}
}
  • D. Dolgopyat
  • Published 1 March 1998
  • Mathematics
  • Annals of Mathematics
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… 
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References

SHOWING 1-10 OF 26 REFERENCES
GIBBS MEASURES IN ERGODIC THEORY
In this paper we introduce the concept of a Gibbs measure, which generalizes the concept of an equilibrium Gibbs distribution in statistical physics. The new concept is important in the study of
Markov approximations and decay of correlations for Anosov flows
We develop Markov approximations for very general suspension flows. Based on this, we obtain a stretched exponential bound on time correlation functions for 3-D Anosov flows that verify ‘uniform
Decay of correlations
*Dedicated to Micheline Ishay I would like to thank P. Boyland, L. Chierchia, V. Donnay, G. De Martino, C. Gole, J. L. Lebowitz, M. Lyubich, M. Rychlik, I. G. Schwarz, S. Vaienti and especially G.
An analogue of the prime number theorem for closed orbits of Axiom A flows
For an Axiom A flow restricted to a basic set we extend the zeta function to an open set containing W(s) > h where h is the topological entropy. This enables us to give an asymptotic formula for the
Markov partitions for anosov flows onn-dimensional manifolds
In this work we construct a Markov partition for transitive Anosov flows, such that the measure of the boundary of the partition is zero. Symbolic dynamics for these flows is also developed.
Zeta functions and the periodic orbit structure of hyperbolic dynamics
© Société mathématique de France, 1990, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les
Prevalence of rapid mixing in hyperbolic flows
We provide necessary and sufficient conditions for a suspension flow, over a subshift of finite type, to mix faster than any power of time. Then we show that these conditions are satisfied if the
Unique ergodicity for horocycle foliations
The notion of unique ergodicity is extended to multidimensional foliations, and it is shown that ifg is the strong stable or strong unstable foliation of a topologically mixing basic set Ωκ for an
Resonances for Axiom ${\bf A}$ flows
Etant donne un flot d'axiome A sur M et des fonctions lisses B, C:M→R, on montre que la fonction de correlation en temps ρ BC pour un etat de Gibbsη a une transformee de Fourier ρ^ BC mesomorphe dans
Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms
Gibbs Measures.- General Thermodynamic Formalism.- Axiom a Diffeomorphisms.- Ergodic Theory of Axiom a Diffeomorphisms.
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