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Distribution of Periods of Closed Trajectories in Exponentially Shrinking Intervals

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Abstract

For hyperbolic flows over basic sets we study the asymptotic of the number of closed trajectories γ with periods T γ lying in exponentially shrinking intervals \({(x - e^{-\delta x}, x + e^{-\delta x}), \; \delta > 0, \; x \to + \infty.}\) A general result is established which concerns hyperbolic flows admitting symbolic models whose corresponding Ruelle transfer operators satisfy some spectral estimates. This result applies to a variety of hyperbolic flows on basic sets, in particular to geodesic flows on manifolds of constant negative curvature and to open billiard flows.

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References

  1. Anantharaman N.: Precise counting results for closed orbits of Anosov flows. Ann. Scient. Éc. Norm. Sup. 33, 33–56 (2000)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Bowen R., Ruelle D.: The ergodic theory of Axiom A flows. Invent. Math 29, 181–202 (1975)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Denker M., Philipp W.: Approximation by Brownian motion for Gibbs measures and flows under a function. Ergod. Th. & Dyn. Sys 4, 541–552 (1984)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Eberlein P.: Geodesic flows on manifolds of non-positive curvature. Proc. Symp. in Pure Math 69, 525–571 (2001)

    MathSciNet  Google Scholar 

  7. Hasselblatt B.: Regularity of the Anosov splitting and of horospheric foliations. Ergod. Th. & Dyn. Sys 14, 645–666 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Hirsch M., Pugh C.: Smoothness of horocycle foliations. J. Diff. Geom 10, 225–238 (1975)

    MathSciNet  MATH  Google Scholar 

  9. Ikawa M.: Decay of solutions of the wave equation in the exterior of several convex bodies. Ann. Inst. Fourier 2, 113–146 (1988)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  11. Lalley, S.: Ruelle’s Perron-Frobenius theorem and the central limit theorem for additive functionals of one-dimensional Gibbs states. In: Adaptive statistical procedures and related topics (Upton, N.Y., 1985), IMS Lecture Notes Monogr. Ser., 8, Hayward, CA: Inst. Math. Statist., 1986, pp. 428–446

  12. Lalley St.: Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits. Acta Math. 163, 1–55 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Margulis G.: On some applications of ergodic theory to the study of manifolds of negative curvature. Func. Anal. App 3, 89–90 (1969)

    MathSciNet  Google Scholar 

  14. Naud F.: Expanding maps on Cantor sets and analytic continuation of zeta functions. Ann. Sci. Ecole Norm. Sup. 38, 116–153 (2005)

    MathSciNet  MATH  Google Scholar 

  15. Parry W., Pollicott M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Asterisque 187-188, 1–268 (1990)

    MathSciNet  Google Scholar 

  16. Petkov V., Stoyanov L.: Correlaitons for pairs of periodic trajectories for open billiards. Nonlinearity 22, 2657–2679 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Pollicott M., Sharp R.: Rates of recurrence for \({\mathbb Z^p}\) and \({\mathbb R^q}\) extensions of subshifts of finite type. J. London Math. Soc 49, 401–416 (1994)

    MathSciNet  MATH  Google Scholar 

  18. Pollicott M., Sharp R.: Exponential error terms for growth functions of negatively curved surfaces. Amer. J. Math. 120, 1019–1042 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pollicott M., Sharp R.: Errors terms for closed orbits of hyperbolic flows. Ergod. Th. & Dyn. Sys. 21, 545–562 (2001)

    MathSciNet  MATH  Google Scholar 

  20. Pollicott M., Sharp R.: Asymptotic expansions for closed orbits in homology classes. Geom. Dedicata 87, 123–160 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pollicott, M., Sharp, R.: Distribution of ergodic sums for hyperbolic maps, Representation theory, dynamical systems, and asymptotic combinatorics. Amer. Math. Soc. Transl. Ser. 2, 217, Providence, RI: Amer. Math. Soc., 2006, pp. 167–183

  22. Pollicott M., Sharp R.: Large deviations, fluctuations and shrinking intervals. Commun. Math. Phys 290, 321–334 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Pugh, C., Shub, M., Wilkinson, A.: Hölder foliations. Duke Math. J. 86, 517–546 (1997); Correction: Duke Math. J. 105, 105–106 (2000)

    Google Scholar 

  24. Ratcliffe J.G.: Foundations of hyperbolic manifolds. Springer-Verlag, New York (1994)

    MATH  Google Scholar 

  25. Ratner M.: Markov partitions for Anosov flows on n-dimensional manifolds. Israel J. Math 15, 92–114 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ruelle D.: A measure associated with Axiom-A attractors. Amer. J. Math 98, 619–654 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ruelle D.: An extension of the theory of Fredholm determinants. Inst. Hautes Études Sci. Publ. Math. 72, 175–193 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. Stoyanov L.: Exponential instability for a class of dispersing billiards. Ergod. Th. & Dyn. Sys 19, 201–226 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Stoyanov L.: Spectrum of the Ruelle operator and exponential decay of correlation for open billiard flows. Amer. J. Math. 123, 715–759 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Stoyanov L.: Spectra of Ruelle transfer operators for Axiom A flows. Nonlinearity 24, 1089–1120 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Stoyanov L.: Non-integrability of open billiard flows and Dolgopyat type estimates. Ergod. Th. & Dyn. Sys 32, 295–311 (2012)

    Article  Google Scholar 

  32. Stoyanov, L.: Ruelle zeta functions and spectra of transfer operators for some Axiom A flows. Preprint 2005, unpublished, available at http://school.maths.uwa.edu.au/~stoyanov/geodAAA.pdf

  33. Walters P.: An introduction to ergodic theory. Springer-Verlag, Berlin (1982)

    Book  MATH  Google Scholar 

  34. Wright, P.: On Ruelle’s lemma and Ruelle zeta functions. http://arxiv.org/abs/1010.4607v1 [math.DS], 2010

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Correspondence to Vesselin Petkov.

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Communicated by S. Zelditch

The first author was partially supported by the ANR project NONAA.

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Petkov, V., Stoyanov, L. Distribution of Periods of Closed Trajectories in Exponentially Shrinking Intervals. Commun. Math. Phys. 310, 675–704 (2012). https://doi.org/10.1007/s00220-012-1419-x

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