Skip to main content
SpringerLink
Log in

Rates of mixing for the Weil–Petersson geodesic flow: exponential mixing in exceptional moduli Spaces

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We establish exponential mixing for the geodesic flow \({\varphi_t\colon T^1S\to T^1S}\) of an incomplete, negatively curved surface S with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil–Petersson flows for the moduli spaces \({\mathcal{M}_{1,1}}\) and \({\mathcal{M}_{0,4}}\) are exponentially mixing, in sharp contrast to the flows for \({\mathcal{M}_{g,n}}\) with \({3g-3+n > 1}\), which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the phase space T 1 S and rescaling the flow \({\varphi_t}\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Araújo V., Butterley O., Varandas P.: Open sets of Axiom A flows with exponentially mixing attractors. Proc. Amer. Math. Soc. 144(7), 2971–2984 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Avila A., Gouëzel S.: Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmller flow. Ann. of Math. 178, 385–442 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Avila A., Gouëzel S., Yoccoz J.-C.: Exponential mixing for the Teichmüller flow. Publ. Math. Inst. Hautes Études Sci. 104, 143–211 (2006)

    Article  MATH  Google Scholar 

  4. V. Araújo and I. Melbourne. Exponential Decay of correlations for nonuniformly hyperbolic flows with a \({C^{1+\alpha}}\) stable foliation, including the classical Lorenz attractor, preprint. arXiv:1504.04316 (2015).

  5. Baladi V., Vallée B.: Exponential decay of correlations for surface semi-flows without finite Markov partitions. Proc. Amer. Math. Soc. 133(3), 865–874 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Baladi, M. Demers and C. Liverani, Exponential Decay of Correlations for Finite Horizon Sinai Billiard Flows, preprint. arXiv:1506.02836 (2015)

  7. K. Burns, H. Masur, C. Matheus, and A. Wilkinson. Rates of mixing for the Weil–Petersson geodesic flow I: no rapid mixing in non-exceptional moduli spaces preprint. arXiv:1312.6012 (2015)

  8. K. Burns, H. Masur, C. Matheus, and A. Wilkinson. Calculating the \({\star}\) connection, preprint, http://math.uchicago.edu/~wilkinso/papers/connection.

  9. Burns K., Masur H., Wilkinson A.: The Weil–Petersson geodesic flow is ergodic. Ann. of Math. (2) 175(2), 835–908 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. M. do Carmo. Riemannian geometry. Birkhuser Boston, Inc., Boston, MA, (1992)

  11. Chernov N.: Markov approximations and decay of correlations for Anosov flows. Ann. of Math. 147, 269–324 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chernov N.: Decay of correlations and dispersing billiards. J. Statist. Phys. 94(3–4), 513–556 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dolgopyat D.: On decay of correlations in Anosov flows. Ann. of Math. (2) 147(2), 357–390 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dolgopyat D.: Prevalence of rapid mixing in hyperbolic flows. Ergodic Theory Dynam. Systems 18(5), 1097–1114 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. U. Hamenstädt. Teichmueller flow and Weil–Petersson flow, preprint. arXiv:1505.01113 (2015)

  16. Heintze E., Im Hof C.-H.: Geometry of horospheres. J. Differential Geom. 12, 481–491 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kowalski O.: Curvature of the Induced Riemannian Metric on the Tangent Bundle of a Riemannian Manifold. J. Reine Angew. Math. 250, 124–129 (1971)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. McMullen C.T.: The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. of Math. 151(1), 327–357 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. S.A. Wolpert. Equiboundedness of the Weil–Petersson metric. preprint. arXiv:1503.00768 (2015).

  21. Young L.-S.: Statistical properties of dynamical systems with some hyperbolicity. Ann. of Math. 147(3), 585–650 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208-2730, USA

    Keith Burns

  2. Department of Mathematics, University of Chicago, 5734 S. University, Chicago, IL, 60637, USA

    Howard Masur & Amie Wilkinson

  3. LAGA, CNRS (UMR 7539), Université Paris 13, Sorbonne Paris Cité, 93439, Villetaneuse, France

    Carlos Matheus

Authors
  1. Keith Burns
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Howard Masur
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Carlos Matheus
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Amie Wilkinson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Amie Wilkinson.

Additional information

K.B. was supported by NSF Grant DMS-1001959, H.M. was supported by NSF Grant DMS 1205016, C.M. was supported by ANR Grant “GeoDyM” (ANR-11-BS01-0004), and A.W. was supported by NSF Grant DMS-1316534.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Burns, K., Masur, H., Matheus, C. et al. Rates of mixing for the Weil–Petersson geodesic flow: exponential mixing in exceptional moduli Spaces. Geom. Funct. Anal. 27, 240–288 (2017). https://doi.org/10.1007/s00039-017-0401-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-017-0401-3

Search

Navigation