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Fluctuations of Time Averages Around Closed Geodesics in Non-Positive Curvature

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Abstract

We consider the geodesic flow for a rank one non-positive curvature closed manifold. We prove an asymptotic version of the Central Limit Theorem for families of measures constructed from regular closed geodesics converging to the Bowen-Margulis-Knieper measure of maximal entropy. The technique expands on ideas of Denker, Senti and Zhang, who proved this type of asymptotic Lindeberg Central Limit Theorem on periodic orbits for expansive maps with the specification property. We extend these techniques from the uniform to the non-uniform setting, and from discrete-time to continuous-time. We consider Hölder observables subject only to the Lindeberg condition and a weak positive variance condition. If we assume a natural strengthened positive variance condition, the Lindeberg condition is always satisfied. Our results extend to dynamical arrays of Hölder observables, and to weighted periodic orbit measures which converge to a unique equilibrium state.

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References

  1. Bálint, P., Melbourne, I.: Statistical properties for flows with unbounded roof function, including the Lorenz attractor. J. Stat. Phys. 172, 1101–1126 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  2. Ballmann, W.: Lectures on spaces of nonpositive curvature, vol. 25 of DMV Seminar, Birkhäuser Verlag, Basel, 1995. With an appendix by Misha Brin

  3. Burns, K., Climenhaga, V., Fisher, T., Thompson, D.J.: Unique equilibrium states for geodesic flows in nonpositive curvature. Geom. Funct. Anal. 28, 1209–1259 (2018)

    Article  MathSciNet  Google Scholar 

  4. Call, B., Thompson, D. J.: Equilibrium states for products of flows and the mixing properties of rank 1 geodesic flows, Preprint arXiv:1906.09315, (2019)

  5. Chazottes, J.-R.: Fluctuations of observables in dynamical systems: from limit theorems to concentration inequalities. In:González-Aguilar, Hernán, Ugalde, Edgardo (eds.) Nonlinear dynamics new directions, of Nonlinear Syst. Complex., vol. 11, pp. 47–85. Springer, Cham (2015)

  6. Climhenhaga, V., Thompson, D.: Unique equilibrium states for flows and homeomorphisms with non-uniform structure. Adv. Math. 303, 745–799 (2016)

    Article  MathSciNet  Google Scholar 

  7. Denker, M., Philipp, W.: Approximation by Brownian motion for Gibbs measures and flows under a function. Ergodic Theory Dynam. Syst. 4, 541–552 (1984)

    Article  MathSciNet  Google Scholar 

  8. Denker, M., Senti, S., Zhang, X.: Fluctuations of ergodic sums on periodic orbits under specification. Dis. Contin. Dyn. Syst. 40, 4665–4687 (2020)

    Article  MathSciNet  Google Scholar 

  9. Dolgopyat, D., Dong, C., Kanigowski, A., Nándori, P.: Flexibility of statistical properties for smooth systems satisfying the central limit theorem, Preprint arXiv:2006.02191, (2020)

  10. Eberlein, P.: Geodesic flows in manifolds of nonpositive curvature, in Smooth ergodic theory and its applications (Seattle, WA, 1999). In: Proc. Sympos. Pure Math., Amer. Math. Soc. Providence, RI, Seattle, WA 69, 525–571 (1999)

  11. Feller, W.: An introduction to probability theory and its applications, vol. 1, 3rd edn. Wiley, New York (1968)

    MATH  Google Scholar 

  12. Gekhtman, I., Taylor, S.J., Tiozzo, G.: A central limit theorem for random closed geodesics: proof of the Chas-Li-Maskit conjecture. Adv. Math. 358, 106852 (2019)

    Article  MathSciNet  Google Scholar 

  13. Gerber, M., Wilkinson, A.: Hölder regularity of horocycle foliations. J. Differ. Geom. 52, 41–72 (1999)

    Article  Google Scholar 

  14. Gouëzel, S.: Limit theorems in dynamical systems using the spectral method, in Hyperbolic dynamics, fluctuations and large deviations. In: Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, Vol. 89, pp. 161–193, 2015

  15. Ibragimov, I.A.: Some limit theorems for stationary processes. Theory Probab. Appl. 7, 349–382 (1962)

    Article  MathSciNet  Google Scholar 

  16. Knieper, G.: The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds. Ann. Math. 148, 291–314 (1998)

    Article  MathSciNet  Google Scholar 

  17. Melbourne, I., Török, A.: Statistical limit theorems for suspension flows. Israel J. Math. 144, 191–209 (2004)

    Article  MathSciNet  Google Scholar 

  18. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Asterisque, no. 187–188, Soc. Math. France (1990)

  19. Pollicott, M.: Estimating variance for expanding maps. http://homepages.warwick.ac.uk/masdbl/preprints

  20. Ratner, M.: The central limit theorem for geodesic flows on \(n\)-dimensional manifolds of negative curvature. Israel J. Math. 16, 181–197 (1973)

    Article  MathSciNet  Google Scholar 

  21. Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics, vol. 79. Springer, Berlin (1982)

    Book  Google Scholar 

  22. Wang, T.: Unique equilibrium states, large deviations and Lyapunov spectra for the Katok Map, arXiv:1903.02677v3, (2019). To appear in Ergodic Theory and Dynamical Systems

  23. Wang, T.: Thermodynamic formalism, statistical properties and multifractal analysis of non-uniformly hyperbolic systems, PhD thesis, The Ohio State University, 2021

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Daniel J. Thompson & Tianyu Wang

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  1. Daniel J. Thompson
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  2. Tianyu Wang
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Correspondence to Daniel J. Thompson.

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Communicated by C.Liverani

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This work is partially supported by NSF grants DMS-1461163 and DMS-1954463.

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Thompson, D.J., Wang, T. Fluctuations of Time Averages Around Closed Geodesics in Non-Positive Curvature. Commun. Math. Phys. 385, 1213–1243 (2021). https://doi.org/10.1007/s00220-021-04062-6

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