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On the Support of Pollicott–Ruelle Resonanant States for Anosov Flows

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Abstract

We show that all generalized Pollicott–Ruelle resonant states of a topologically transitive C ∞ Anosov flow with an arbitrary C ∞ potential have full support.

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References

  1. Anosov D.V.: Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Matematicheskogo Instituta im. VA Steklova 90, 3–210 (1967)

    MathSciNet  Google Scholar 

  2. Blank M., Keller G., Liverani C.: Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15, 1905 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Bowen R.: Markov partitions for Axiom A diffeomorphisms. Am. J. Math. 92(3), 725–747 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bowen R.: Symbolic dynamics for hyperbolic flows. Am. J. Math. 95(2), 429–460 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. Butterley O., Liverani C.: Smooth Anosov flows: correlation spectra and stability. J. Modern Dyn. 1(2), 301–322 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Barreira L., Pesin Y.: Nonuniform Hyperbolicity, Dynamics of Systems with Nonzero Lyapunov Exponents. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  7. Brin M., Stuck G.: Introduction to Dynamical Systems. Cambridge University Press, Cambridge (2002)

    Book  MATH  Google Scholar 

  8. Baladi V., Tsujii M.: Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier 57(1), 127–154 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dyatlov S., Faure F., Guillarmou C.: Power spectrum of the geodesic flow on hyperbolic manifolds. Anal. PDE 8(4), 923–1000 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Dyatlov, S., Guillarmou, C.: Pollicott–Ruelle resonances for open systems. Ann. Henri Poincaré (2016). doi:10.1007/s00023-016-0491-8

  11. Dolgopyat D.: On decay of correlations in Anosov flows. Ann. Math. 147, 357–390 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dyatlov, S., Zworski, M.: Dynamical zeta functions for Anosov flows via microlocal analysis. Annal. de l’ENS 49(3), 543–577 (2016)

  13. Faure F., Roy N.: Ruelle–Pollicott resonances for real analytic hyperbolic maps. Nonlinearity 19(6), 1233–1252 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Faure, F., Roy, N., Sjöstrand, J.: Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances. Open Math. J. 1, 35–81 (2008)

  15. Faure F., Sjöstrand J.: Upper bound on the density of Ruelle resonances for Anosov flows. Commun. Math. Phys. 308(2), 325–364 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Gouëzel S., Liverani C.: Banach spaces adapted to Anosov systems. Ergodic Theory Dyn. Syst. 26(1), 189–217 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gouëzel S., Liverani C.: Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differ. Geom. 79(3), 433–477 (2008)

    MathSciNet  MATH  Google Scholar 

  18. Giulietti P., Liverani C., Pollicott M.: Anosov flows and dynamical zeta functions. Ann. Math. (2) 178(2), 687–773 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hasselblatt, B.: Hyperbolic Dynamical Systems. Handbook of Dynamical Systems, vol. 1A. North-Holland, Amsterdam, pp. 239–319 (2002)

  20. Hörmander, L.: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Reprint of the second (1990) edition. Springer, Berlin (2003)

  21. Hirsch, M.W., Pugh, C.C.: Stable Manifolds and Hyperbolic Sets. Global Analysis (Proc. Sympos. Pure Math., vol. XIV, Berkeley, Calif., 1968), pp. 133–163, (1970)

  22. Hirsch M.W., Pugh C.C., Shub M.: Invariant manifolds. Bull. Am. Math. Soc. 76, 1015–1019 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kitaev A.Yu.: Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness. Nonlinearity 12(1), 141–179 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Liverani C.: On contact Anosov flows. Ann. Math. (2) 159(3), 1275–1312 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nonnenmacher S., Zworski M.: Decay of correlations for normally hyperbolic trapping. Invent. Math. 200(2), 345–438 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  26. Plante J.F.: Anosov flows. Am. J. Math. 94, 729–754 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pugh C., Shub M.: The Ω-stability theorem for flows. Invent. Math. 11, 150–158 (1970)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  28. Rugh H.H.: The correlation spectrum for hyperbolic analytic maps. Nonlinearity 5(6), 1237 (1992)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Tsujii M.: Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity 23(7), 1495 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Young L.S.: What are SRB measures, and which dynamical systems have them?. J. Stat. Phys. 108(5), 733–754 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Universität Paderborn, Paderborn, Germany

    Tobias Weich

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  1. Tobias Weich
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Correspondence to Tobias Weich.

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Communicated by Dmitry Dolgopyat.

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Weich, T. On the Support of Pollicott–Ruelle Resonanant States for Anosov Flows. Ann. Henri Poincaré 18, 37–52 (2017). https://doi.org/10.1007/s00023-016-0514-5

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  • DOI: https://doi.org/10.1007/s00023-016-0514-5

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