This event is part of the activities around the chaire Jean-Morlet of Boris Hasselblatt. Among the topics scheduled in this event, there is a mini-course by Keith Burns and myself around the dynamics of the Weil-Petersson (WP) geodesic flow.
In our mini-course, Keith and I plan to cover some aspects of Burns-Masur-Wilkinson theorem on the ergodicity of WP flow and, maybe, some points of our joint work with Masur and Wilkinson on the rates of mixing of WP flow.
In order to help me prepare my talks, I thought it could be a good idea to make my notes available on this blog.
So, this post starts a series of 6 posts (vaguely corresponding the 6 lectures of the mini-course) on the dynamics of the WP flow.
The WP flow and its close cousin the Teichmüller flow are studied in the literature in part because its dynamical properties allow to understand certain geometrical aspects of Riemann surfaces.
The precise definitions of these flows will be given later, but, for now, let us list some of their properties.
|Teichmüller flow||WP flow|
|(a)||comes from a Finsler||comes from a Riemannian metric|
|(c)||is part of an action||is not part of an action|
|(d)||non-uniformly hyperbolic||singular hyperbolic|
|(e)||related to flat geometry of curves||related to hyperbolic geometry of curves|
|(g)||periodic orbits are dense||periodic orbits are dense|
|(h)||finite topological entropy||infinite topological entropy|
|(i)||ergodic for the Liouville measure||ergodic for the Liouville measure|
|(j)||metric entropy||metric entropy|
|(k)||exponential rate of mixing||mixing at most polynomial (in genus )|
The items above serve to highlight some differences between the Teichmüller and WP flows.
In fact, the Teichmüller flow is associated to a Finsler, i.e., a continuous family of norms, on the fibers of the cotangent bundle of the moduli spaces (actually, a but not family of norms [see pages 308 and 309 of Hubbard's book]), while the WP flow is associated to a Riemannian and, actually, Kähler, metric. We will come back to this point later when defining the WP metric.
In particular, the item (a) says that the WP flow comes from a metric that is richer than the metric generating the Teichmüller flow.
On the other hand, the item (b) says that the WP geodesic flow has a not so nice dynamics because it is incomplete, that is, there are certain WP geodesics that “leave”/“go to infinity” in finite time. In particular, the WP flow is not defined for all time when we start from certain initial datum. We will make more comments on this later. Nevertheless, Wolpert showed that the WP flow is defined for all time for almost every initial data (with respect to the Liouville [volume] measure induced by WP metric), and, thus, the WP flow is a legitim flow from the point of view of Dynamics/Ergodic Theory.
The item (c) says that WP flow differs from Teichmüller flow because the former is not part of a -action while the latter corresponds to the action of the diagonal subgroup of a natural -action on the unit cotangent bundle of the moduli spaces of curves. Here, it is worth to mention that the mere fact that the Teichmüller flow is part of a -action makes its dynamics very rich: for instance, once one shows that the Teichmüller flow is ergodic (with respect to some -invariant probability measure), it is possible to apply Howe-Moore’s theorem (or variants of it) to improve ergodicity into mixing (and, actually, exponential mixing) of Teichmüller flow (see e.g. this post for more details).
The item (d) says that both WP and Teichmüller flows are non-uniformly hyperbolic (in the sense of Pesin theory), but they are so for distinct reasons. The non-uniform hyperbolicity of the Teichmüller flow was shown by Veech (for the “volume”/Masur-Veech measure) and Forni (for an arbitrary Teichmüller flow invariant probability measure) and it follows from uniform estimates for the derivative of the Teichmüller flow on bounded sets. On the other hand, the non-uniform hyperbolicity of the WP flow requires a slightly different argument because the curvatures of WP metric might approach or at certain places near the “boundary” of the moduli spaces. We will return to this point in the future.
The item (e) says that, concerning applications of these flows to the investigation of curves/Riemann surfaces, it is natural to study the Teichmüller flow whenever one is interested in the properties of flat metrics with conical singularities (cf. this post here), while it is more natural to study the WP metric/flow whenever one is interested in the properties of hyperbolic metrics: for instance, Wolpert showed that the hyperbolic length of a closed geodesic in a fixed free homotopy class is a convex function along orbits of the WP flow, Mirzakhani proved that the growth of the hyperbolic lengths of simple geodesics on hyperbolic surfaces is related to the WP volume of the moduli space, and, after the works of Bridgeman, McMullen and more recently Bridgeman, Canary, Labourie and Sambarino (among other authors), we know that the Weil-Petersson metric is intimately related to thermodynamical invariants (entropy, pressure, etc.) of the geodesic flow on hyperbolic surfaces.
Concerning items (f) to (h), Pollicott-Weiss-Wolpert showed the transitivity and denseness of periodic orbits of the WP flow in the particular case of the unit cotangent bundle of the moduli space (of once-punctured tori). In general, the transitivity, the denseness of periodic orbits and the infinitude of the topological entropy of the WP flow on the unit cotangent bundle of (for any , ) were shown by Brock-Masur-Minsky. Moreover, Hamenstädt proved the ergodic version of the denseness of periodic orbits, i.e., the denseness of the subset of ergodic probability measures supported on periodic orbits in the set of all ergodic WP flow invariant probability measures.
The ergodicity of WP flow (mentioned in item (i)) was first studied by Pollicott-Weiss in the particular case of the unit cotangent bundle of the moduli space of once-punctured tori: they showed that if the first two derivatives of the WP flow on are suitably bounded, then this flow is ergodic. More recently, Burns-Masur-Wilkinson were able to control in general the first derivatives of WP flow and they used their estimates to show the following theorem:
Theorem 1 (Burns-Masur-Wilkinson) The WP flow on the unit cotangent bundle of is ergodic (for any , ) with respect to the Liouville measure of the WP metric. Actually, it is Bernoulli (i.e., it is measurably isomorphic to a Bernoulli shift) and, a fortiori, mixing. Furthermore, its metric entropy is positive and finite.
A detailed explanation of this theorem will occupy the next four posts of this series. For now, we will just try to describe the general lines of Burns-Masur-Wilkinson arguments in Section 1 below.
However, before passing to this subject, let us make some comments about item (k) above on the rate of mixing of Teichmüller and WP flows.
Generally speaking, it is expected that the rate of mixing (decay of correlations) of a system (diffeomorphism or flow) displaying a “reasonable” amount of hyperbolicity is exponential: for example, the property of exponential rate of mixing was shown by Dolgopyat (see also this article of Liverani and this blog post) for contact Anosov flows (such as geodesic flows on compact Riemannian manifolds with negative curvature), and by Avila-Gouëzel-Yoccoz and Avila-Gouëzel for the Teichmüller flow equipped with “nice” measures.
Here, we recall that the rate of mixing/decay of correlations of a (mixing) flow is the speed of convergence of the correlations functions to as (for “reasonably smooth” observables and ), that is, the speed of to mix distinct regions of the phase space (such as the supports of the observables and ).
In this context, given the ergodicity and mixing theorem of Burns-Masur-Wilkinson, it is natural to try to “determine” the rate of mixing of WP flow. In this direction, we obtained the following result (in a preprint still in preparation):
- at most polynomial for and
- rapid (super-polynomial) for , .
We will present a sketch of proof of this result in the last post of this series. For now, we will content ourselves with a vague description of the geometrical reason for the difference in the rate of mixing of the Teichmüller and WP flows in Section 2 below.
Closing this introduction, let us give a plan of this series of posts. Firstly, we will complete today’s post by discussing the general scheme for the proof of Burns-Masur-Wilkinson theorem (ergodicity of WP flow) in Section 1 below and by explaining the geometry behind the rate of mixing of WP flow in Section 2. Then, in the second post of this series, we will define the WP geodesic flow on the unit cotangent bundle of the moduli spaces of curves and we will “reduce” Burns-Masur-Wilkinson theorem to the verification of adequate estimates of the derivatives of WP flow via a certain ergodicity criterion à la Katok-Strelcyn. After that, we will spend the third and fourth post discussing the proof of the ergodicity criterion à la Katok-Strelcyn, and we will dedicate the fifth post to show that the WP geodesic flow satisfies all assumptions of the ergodicity criterion. Finally, the last post will concern the rates of mixing of WP flow.