## Journal of the Mathematical Society of Japan

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

Masato TSUJII

#### Abstract

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.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 2 (2018), 757-821.

Dates
First available in Project Euclid: 18 April 2018

https://projecteuclid.org/euclid.jmsj/1524038673

Digital Object Identifier
doi:10.2969/jmsj/07027595

Mathematical Reviews number (MathSciNet)
MR3787739

Zentralblatt MATH identifier
06902441

#### Citation

TSUJII, Masato. Exponential mixing for generic volume-preserving Anosov flows in dimension three. J. Math. Soc. Japan 70 (2018), no. 2, 757--821. doi:10.2969/jmsj/07027595. https://projecteuclid.org/euclid.jmsj/1524038673

#### References

• V. Baladi, M. F. Demers and C. Liverani, Exponential decay of correlations for finite horizon Sinai billiard flows, Invent. Math., 211 (2018), 39–177. \enlargethispage\baselineskip
• V. Baladi and C. Liverani, Exponential decay of correlations for piecewise cone hyperbolic contact flows, Comm. Math. Phys., 314 (2012), 689–773.
• V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble), 57 (2007), 127–154.
• N. I. Chernov, Markov approximations and decay of correlations for Anosov flows, Ann. of Math. (2), 147 (1998), 269–324.
• F. den Hollander, Large deviations, Fields Institute Monographs, 14, American Mathematical Society, Providence, RI, 2000.
• D. Dolgopyat, On decay of correlations in Anosov flows, Ann. of Math. (2), 147 (1998), 357–390.
• D. Dolgopyat, Prevalence of rapid mixing in hyperbolic flows, Ergodic Theory Dynam. Systems, 18 (1998), 1097–1114.
• D. Dolgopyat, Prevalence of rapid mixing, II, Topological prevalence, Ergodic Theory Dynam. Systems, 20 (2000), 1045–1059.
• F. Faure and M. Tsujii, The semiclassical zeta function for geodesic flows on negatively curved manifolds, Invent. Math., 208 (2017), 851–998.
• F. Faure and M. Tsujii, Prequantum transfer operator for symplectic Anosov diffeomorphism, Astérisque, 375 (2015), ix+222 pages.
• M. Field, I. Melbourne and A. Török, Stability of mixing and rapid mixing for hyperbolic flows, Ann. of Math. (2), 166 (2007), 269–291.
• S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189–217.
• A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.
• G. Keller and C. Liverani, Stability of the spectrum for transfer operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 141–152.
• C. Liverani, On contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275–1312.
• A. Martinez, An introduction to semiclassical and microlocal analysis, Universitext. Springer-Verlag, New York, 2002.
• J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286–294.
• F. Nicola and L. Rodino, Global pseudo-differential calculus on Euclidean spaces, Pseudo-Differential Operators, Birkhauser, 2010.
• M. E. Taylor, Pseudodifferential operators and nonlinear PDE, Progress in Mathematics, 100, Birkhäuser Boston Inc., Boston, MA, 1991.
• M. Tsujii, A measure on the space of smooth mappings and dynamical system theory, J. Math. Soc. Japan, 44 (1992), 415–425.
• M. Tsujii, Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, 23 (2010), 1495–1545.
• M. Tsujii, Contact Anosov flows and the Fourier–Bros–Iagolnitzer transform, Ergodic Theory Dynam. Systems, 32 (2012), 2083–2118.