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Dynamical determinants for smooth hyperbolic dynamics

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Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

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Abstract

As in Chapters 4 and 5, we consider a diffeomorphism on a hyperbolic basic set and a differentiable weight. In this chapter, we study the associated weighted dynamical determinant, giving a lower bound on the disc in which this determinant is analytic and where its zeroes admit a spectral interpretation. We apply the results obtained on the weighted dynamical determinant to study the dynamical zeta function.

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Notes

  1. 1.

    Or an \(C^{r}\) Anosov diffeomorphism on a connected manifold \(M\), not necessarily transitive.

  2. 2.

    This is the same definition as (3.4) in the expanding case, except that \(|\det(\mathrm{Id}-DT^{-m}(x))|\) there is replaced by \(|\det(\mathrm{Id}-DT^{m}(x))|\) here. This is because the transfer operator \(\mathcal{L}_{g}\) is now defined by forward composition with \(T\).

  3. 3.

    By Lemma A.3 and the perturbation results from §5.3 one can construct examples of \(C^{r}\) hyperbolic diffeomorphisms with nontrivial resonances for \(r<\infty\).

  4. 4.

    See e.g. [178] or [105] for a definition of expansive and specification. Beware that expansive is not the same as expanding.

  5. 5.

    See (6.40) below for a more precise estimate.

  6. 6.

    As usual, if \(\inf|g||_{\Lambda}=0\) we approach \(|g|\) by non-vanishing functions. See Appendix B

  7. 7.

    The decomposition is independent of \(t\) and \(s\).

  8. 8.

    This choice is not essential in the proof of Proposition 6.9, but it will be important to prove Proposition 6.11.

  9. 9.

    See also [31, App C] and Footnote 28 of Chapter 4.

  10. 10.

    Or see Footnote 19.

  11. 11.

    See also [31, App C] and Footnote 28 of Chapter 4.

  12. 12.

    To write a formal proof involving charts, we may integrate by parts as many times as we like with respect to \(x\) in the relevant kernels, as in the second step of the proof of Lemma 6.9.

  13. 13.

    See also [31, App C] and Footnote 28 of Chapter 4.

  14. 14.

    Note that \(\mathrm{tr}\, \,\mathbb{T}_{x}^{m}\ne0\) only if \(x\in V\). If \(T^{m}(x)=x\) this implies \(x\in\Lambda\).

  15. 15.

    For the sake of comparison with [88], note that their transfer operator is defined by composing with \(T^{-1}\) so \(E^{s}\) there replaces \(E^{u}\) here.

  16. 16.

    See [73, 137] or [80].

  17. 17.

    Kitaev worked in the slightly more general setting of Mixed Transfer Operators.

  18. 18.

    In this respect, [68, Remark 2 after Thm 5] should be taken with a grain of salt.

References

  1. Adam, A.: Generic non-trivial resonances for Anosov diffeomorphisms. Nonlinearity 30, 1146–1164 (2017)

    Article  MathSciNet  Google Scholar 

  2. Baillif, M.: Kneading operators, sharp determinants, and weighted Lefschetz zeta functions in higher dimensions. Duke Math. J. 124, 145–175 (2004)

    Article  MathSciNet  Google Scholar 

  3. Baladi, V.: Periodic orbits and dynamical spectra. Ergodic Theory Dynam. Systems 18, 255–292 (1998)

    Article  MathSciNet  Google Scholar 

  4. Baladi, V.: The quest for the ultimate anisotropic Banach space. J. Stat. Phys. Special Volume for D. Ruelle and Ya. Sinai 166, 525–557 (2017)

    MathSciNet  MATH  Google Scholar 

  5. Baladi, V., Kitaev, A., Ruelle, D., Semmes, S.: Sharp determinants and kneading operators for holomorphic maps. Tr. Mat. Inst. Steklova 216, Din. Sist. i Smezhnye Vopr., 193–235 (1997); translation in Proc. Steklov Inst. Math. 216, 186–228 (1997)

    Google Scholar 

  6. Baladi, V., Ruelle, D., Sharp determinants. Invent. Math. 123, 553–574 (1996)

    Article  MathSciNet  Google Scholar 

  7. Baladi, V., Tsujii, M.: Dynamical determinants for hyperbolic diffeomorphisms via dyadic decomposition, unpublished manuscript (2005)

    Google Scholar 

  8. Baladi, V., Tsujii, M.: Dynamical determinants and spectrum for hyperbolic diffeomorphisms. In: Burns, K., Dolgopyat, D., Pesin, Ya. (eds.) Probabilistic and Geometric Structures in Dynamics, pp. 29–68, Contemp. Math., 469, Amer. Math. Soc., Providence, RI (2008)

    Chapter  Google Scholar 

  9. Bowen, R.: Some systems with unique equilibrium states. Math. Systems Theory 8, 193–202 (1974–1975).

    Article  MathSciNet  Google Scholar 

  10. Dang, N.V., Rivière, G.: Spectral analysis of Morse-Smale gradient flows. arXiv:1605.05516

  11. Dang, N.V., Rivière, G.: Pollicott-Ruelle spectrum and Witten Laplacians. arXiv:1709.04265

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  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)

    Article  MathSciNet  Google Scholar 

  15. Faure, F., Tsujii, M.: Prequantum transfer operator for symplectic Anosov diffeomorphism. Astérisque No. 375. Soc. Math. France, Paris (2015)

    MATH  Google Scholar 

  16. Faure, F., Tsujii, M.: Band structure of the Ruelle spectrum of contact Anosov flows. C.R. Acad. Sci. Paris, Ser. I. 351, 385–391 (2013)

    Article  MathSciNet  Google Scholar 

  17. Faure, F., Tsujii, M.: The semiclassical zeta function for geodesic flows on negatively curved manifolds. Invent. Math. 208, 851–998 (2017)

    Article  MathSciNet  Google Scholar 

  18. Fried, D.: The zeta functions of Ruelle and Selberg I. Ann. Sci. École Norm. Sup. (4) 19, 491–517 (1986)

    Article  MathSciNet  Google Scholar 

  19. Fried, D.: Meromorphic zeta functions for analytic flows. Comm. Math. Phys. 174, 161–190 (1995)

    Article  MathSciNet  Google Scholar 

  20. Fried, D.: The flat-trace asymptotics of a uniform system of contractions. Ergodic Theory Dynam. Systems 15, 1061–1073 (1995)

    Article  MathSciNet  Google Scholar 

  21. Giulietti, P., Liverani, C., Pollicott, M.: Anosov flows and dynamical zeta functions. Annals of Mathematics 178, 687–773 (2013)

    Article  MathSciNet  Google Scholar 

  22. Gouëzel, S., Liverani, C.: Banach spaces adapted to Anosov systems. Ergodic Theory Dynam. Systems 26, 189–217 (2006)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Grothendieck, A.: La théorie de Fredholm. Bull. Soc. Math. France 84, 319–384 (1956)

    Article  MathSciNet  Google Scholar 

  25. Hille, E: Analytic function theory. Vol. 1. Introduction to Higher Mathematics, Ginn and Company, Boston (1959)

    MATH  Google Scholar 

  26. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  27. Kitaev, A.Yu.: Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness. Nonlinearity 12, 141–179 (1999). Corrigendum: “Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness”. Nonlinearity 12, 1717–1719 (1999)

    Article  MathSciNet  Google Scholar 

  28. Liverani, C.: Fredholm determinants, Anosov maps and Ruelle resonances. Discrete Contin. Dyn. Syst. 13, 1203–1215 (2005)

    Article  MathSciNet  Google Scholar 

  29. Liverani, C., Tsujii, M.: Zeta functions and dynamical systems. Nonlinearity 19, 2467–2473 (2006)

    Article  MathSciNet  Google Scholar 

  30. Mayer, D.: The Ruelle–Araki Transfer Operator in Classical Statistical Mechanics. Lecture Notes in Phys. 123, Springer-Verlag, Berlin-New York (1980)

    MATH  Google Scholar 

  31. Naud, F.: Anosov diffeomorphisms with non-trivial Ruelle spectrum. Personal communication (June 2015)

    Google Scholar 

  32. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque No. 187–188. Soc. Math. France, Paris (1990)

    MATH  Google Scholar 

  33. Ruelle, D.: Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34, 231–242 (1976)

    Article  MathSciNet  Google Scholar 

  34. Rugh, H.H.: The correlation spectrum for hyperbolic analytic maps. Nonlinearity 5, 1237–1263 (1992)

    Article  MathSciNet  Google Scholar 

  35. Rugh, H.H.: Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems. Ergodic Theory Dynam. Systems 16, 805–819 (1996)

    Article  MathSciNet  Google Scholar 

  36. Slipantschuk, J., Bandtlow, O.F., Just, W.: Analytic expanding circle maps with explicit spectra. Nonlinearity 26, 3231–3245 (2013)

    Article  MathSciNet  Google Scholar 

  37. Slipantschuk, J., Bandtlow, O.F., Just, W.: Complete spectral data for analytic Anosov maps of the torus. Nonlinearity 30, 2667–2686 (2017)

    Article  MathSciNet  Google Scholar 

  38. Tangerman, F.: Meromorphic Continuation of Ruelle Zeta Functions (Heat Operators). Ph.D. thesis, Boston (1986)

    Google Scholar 

  39. Tsujii, M.: Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity 23, 1495–1545 (2010)

    Article  MathSciNet  Google Scholar 

  40. Tsujii, M.: Contact Anosov flows and the Fourier-Bros-Iagolnitzer transform. Ergodic Theory Dynam. Systems 32, 2083–2118 (2012)

    Article  MathSciNet  Google Scholar 

  41. Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin (1982)

    MATH  Google Scholar 

  42. Zworski, M.: Mathematical study of scattering resonances. Bull. Math. Sci. 7, 1–85 (2017)

    Article  MathSciNet  Google Scholar 

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  1. IMJ-PRG, Sorbonne Université and CNRS, Paris, France

    Viviane Baladi

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Baladi, V. (2018). Dynamical determinants for smooth hyperbolic dynamics. In: Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol 68. Springer, Cham. https://doi.org/10.1007/978-3-319-77661-3_6

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