# 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} }

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

- Mathematics
- 2008

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 the… Expand

Markov approximations and decay of correlations for Anosov flows

- Mathematics
- 1998

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… Expand

The semiclassical zeta function for geodesic flows on negatively curved manifolds

- Mathematics
- 2013

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 the… Expand

Exponential Decay of Correlations for Piecewise Cone Hyperbolic Contact Flows

- Mathematics, Physics
- 2011

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 correlations… Expand

Exponential decay of correlations for finite horizon Sinai billiard flows

- Mathematics, Physics
- 2015

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, we… Expand

Global Pseudo-differential Calculus on Euclidean Spaces

- Mathematics
- 2010

Background meterial.- Global Pseudo-Differential Calculus.- ?-Pseudo-Differential Operators and H-Polynomials.- G-Pseudo-Differential Operators.- Spectral Theory.- Non-Commutative Residue and Dixmier… Expand

Stability of mixing and rapid mixing for hyperbolic flows

- Mathematics
- 2007

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… Expand

Banach spaces adapted to Anosov systems

- Mathematics
- Ergodic Theory and Dynamical Systems
- 2005

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 spaces… Expand

Prevalence of rapid mixing in hyperbolic flows

- Mathematics
- 1998

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… Expand

Prequantum transfer operator for symplectic Anosov diffeomorphism

- Physics, Mathematics
- 2012

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 spectral… Expand