Abstract
We describe the proof that the geodesic flow for the Weil–Petersson metric on the moduli space of Riemann surfaces is ergodic and in fact Bernoulli. Other chapters in this volume complement this summary by describing in depth the needed implementation of the Hopf argument and some of the pertinent aspects of moduli spaces of Riemann surfaces (and Teichmüller theory).
This is a preview of subscription content, log in via an institution to check access.
Notes
- 1.
Before our work appeared, Pollicott and Weiss [PW09] gave a fairly complete outline of how to prove ergodicity for the Weil–Petersson metric in these cases. They studied the model case of a negatively curved surface whose singularities coincide with a surface of revolution for a polynomial and proved ergodicity of the geodesic flow in this case. They say that the missing ingredients are the bounds on the first and second derivatives of the geodesic flow.
- 2.
For almost every v ∈ T 1 M, there exists an infinite geodesic (necessarily unique) tangent to v.
References
D.V. Anosov, Geodesic Flows on Closed Riemann Manifolds with Negative Curvature. Trudy Mat. Institute. V.A. Steklova, vol. 90 (American Mathematical Society, Providence, RI, 1969/1967)
W. Ballmann, M. Brin, K. Burns, On the differentiability of horocycles and horocycle foliations. J. Differ. Geom. 26(2), 337–347 (1987)
M. Bridgeman, Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space. Geom. Topol. 14, 799–831 (2010)
J.F. Brock, H. Masur, Y. Minsky, Asymptotics of Weil–Petersson geodesic. I. Ending laminations, recurrence, and flows. Geom. Funct. Anal. 19, 1229–1257 (2010)
K. Burns, H. Masur, A. Wilkinson, The Weil–Petersson geodesic flow is ergodic. Ann. Math. (2) 175(2), 835–908 (2012)
J. Cheeger, D.G. Ebin, Comparison Theorems in Riemannian Geometry (AMS Chelsea Publishing, Providence, RI, 2008); revised reprint of the 1975 original
D. Dolgopyat, H. Hu, Y. Pesin, An example of a smooth hyperbolic measure with countably many ergodic components, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999). Proceedings of Symposia in Pure Mathematics, vol. 69 (American Mathematical Society, Providence, RI, 2001)
P.B. Eberlein, Geometry of Nonpositively Curved Manifolds. Chicago Lectures in Mathematics (University of Chicago Press, Chicago, IL, 1996)
P.B. Eberlein, Geodesic flows in manifolds of nonpositive curvature, in Smooth Ergodic Theory and its Applications (Seattle, WA, 1999). Proceedings of Symposia in Pure Mathematics, vol. 69 (American Mathematical Society, Providence, RI, 2001), pp. 525–571
U. Hamenstädt, Dynamical properties of the Weil–Petersson metric, in In the Tradition of Ahlfors–Bers. V. Contemporary Mathematics, vol. 510 (American Mathematical Society, Providence, RI, 2010), pp. 109–127
B. Hasselblatt, Introduction to hyperbolic dynamics and ergodic theory, in Ergodic Theory and Negative Curvature, ed. by B. Hasselblatt. Lecture Notes in Mathematics, vol. 2164 (Springer, Cham, 2017). doi:10.1007/978-3-319-43059-1_1
E. Hopf, Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber. Verh. Sächs. Akad. Wiss. Leipzig 91, 261–304 (1939)
A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications, vol. 54 (Cambridge University Press, Cambridge, 1995)
A. Katok, J.-M. Strelcyn, F. Ledrappier, F. Przytycki, Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities. Lecture Notes in Mathematics, vol. 1222 (Springer, New York, 1986)
C.T. McMullen, The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. Math. 151, 327–357 (2000)
C.T. McMullen, Thermodynamics, dimension and the Weil–Petersson metric. Invent. Math. 173, 365–425 (2008)
M. Mirzakhani, Growth of the number of simple closed geodesics on hyperbolic surfaces. Ann. Math. 168, 97–125 (2008)
Y.B. Pesin, Characteristic Lyapunov exponents and ergodic properties of smooth dynamical systems with invariant measure. Dokl. Akad. Nauk SSSR 226, 774–777 (1976)
M. Pollicott, H. Weiss, Ergodicity of the geodesic flow on noncomplete negatively curved surfaces. Asian J. Math. 13, 405–419 (2009)
M. Pollicott, H. Weiss, S.A. Wolpert, Topological dynamics of the Weil–Petersson geodesic flow. Adv. Math. 223 (2010), 1225–1235
C.C. Pugh, The C 1+α hypothesis in Pesin theory. Inst. Hautes Études Sci. Publ. Math. 59, 143–161 (1984)
S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds. Tôhoku Math. J. 10, 338–354 (1958)
J.-P. Serre: Rigiditée de foncteur d’Jacobi d’echelonn ≥ 3. 1961, Sem. H. Cartan 1960/1961 Appendix to Exp. 17.
Y.G. Sinai, Dynamical systems with elastic reflections: ergodic properties of scattering billiards. Russ. Math. Surv. 25, 137–189 (1970)
S.A. Wolpert, Understanding Weil–Petersson Curvature. Geometry and Analysis. No. 1. Advanced Lectures in Mathematics (ALM), vol. 17 (International Press, Somerville, MA, 2011), pp. 495–515
S.A. Wolpert, The Fenchel–Nielsen deformation. Ann. Math. 115, 501–528 (1982)
S.A. Wolpert, Geometry of the Weil–Petersson completion of Teichmüller space, in Surveys in Differential Geometry (Boston, MA, 2002), vol. VIII (International Press, Somerville, MA, 2003), pp. 357–393
S.A. Wolpert, Behavior of geodesic-length functions on Teichmüller space. J. Differ. Geom. 79, 277–334 (2008)
S.A. Wolpert, Extension of the Weil–Petersson connection. Duke Math. J. 146, 281–303 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Burns, K., Masur, H., Wilkinson, A. (2017). Ergodicity of the Weil–Petersson Geodesic Flow. In: Hasselblatt, B. (eds) Ergodic Theory and Negative Curvature. Lecture Notes in Mathematics, vol 2164. Springer, Cham. https://doi.org/10.1007/978-3-319-43059-1_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-43059-1_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-43058-4
Online ISBN: 978-3-319-43059-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)