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).
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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.
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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
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