Exponential mixing for generic volume-preserving Anosov flows in dimension three

@article{Tsujii2016ExponentialMF,
  title={Exponential mixing for generic volume-preserving Anosov flows in dimension three},
  author={Masato Tsujii},
  journal={arXiv: Dynamical Systems},
  year={2016}
}
  • M. Tsujii
  • Published 1 January 2016
  • Mathematics
  • arXiv: Dynamical Systems
Let $M$ be a closed $3$-dimensional Riemann manifold and let $3\le r\le \infty$. We prove that there exists an open dense subset in the space of $C^r$ volume-preserving Anosov flows on $M$ such that all the flows in it are exponentially mixing. 

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References

SHOWING 1-10 OF 23 REFERENCES
Quasi-compactness of transfer operators for contact Anosov flows
For any Cr contact Anosov flow with r ≥ 3, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all the Cr functions, such that theExpand
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 ‘uniformExpand
The semiclassical zeta function for geodesic flows on negatively curved manifolds
We consider the semi-classical (or Gutzwiller–Voros) zeta functions for $$C^\infty $$C∞ contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to theExpand
Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows
We prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form, in dimension three. This is the first time exponential decay of correlationsExpand
Exponential decay of correlations for finite horizon Sinai billiard flows
We prove exponential decay of correlations for the billiard flow associated with a two-dimensional finite horizon Lorentz Gas (i.e., the Sinai billiard flow with finite horizon). Along the way, weExpand
Global Pseudo-differential Calculus on Euclidean Spaces
Background meterial.- Global Pseudo-Differential Calculus.- ?-Pseudo-Differential Operators and H-Polynomials.- G-Pseudo-Differential Operators.- Spectral Theory.- Non-Commutative Residue and DixmierExpand
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 2Expand
Banach spaces adapted to Anosov systems
We study the spectral properties of the Ruelle–Perron–Frobenius operator associated to an Anosov map on classes of functions with high smoothness. To this end we construct anisotropic Banach spacesExpand
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 theExpand
Prequantum transfer operator for symplectic Anosov diffeomorphism
We define the prequantization of a symplectic Anosov diffeomorphism f:M-> M, which is a U(1) extension of the diffeomorphism f preserving an associated specific connection, and study the spectralExpand
...
1
2
3
...