This research announcement has 6 pages and it is divided into two parts:
- a) we present a short (complete) proof of a polynomial upper bound for the rate of mixing of Weil-Petersson (WP) flow on the unit cotangent bundles of moduli spaces of surfaces of genus and punctures with , and
- b) we give a sketch of proof of the rapid (i.e., faster than polynomial) mixing property for the Weil-Petersson (WP) flow on the unit cotangent bundles of moduli spaces of four-times punctered spheres and once-punctured torii (that is, in the cases when ).
[Update (April 25, 2015): In view of the recent results of Araujo and Melbourne, Burns, Masur, Wilkinson, Wolpert and I think that we can improve the result in item b) by showing the exponential mixing property for the WP flow on the unit cotangent bundles of and . We hope to provide more details about this in a forthcoming post.]
As we explain in our note, the speed of mixing of the WP flow on the unit cotangent bundle of is polynomial at most when because the (strata of the) boundary of the moduli spaces (in Deligne-Mumford compactification) looks like a non-trivial product of WP metrics on non-trivial moduli spaces of surfaces of lower “complexity” (smaller genus and/or less punctures) when .
More concretely, using this geometrical information on the WP metric near the boundary of , one can produce lots of geodesics spending a lot of time near the boundary of the moduli spaces traveling almost-parallel to one of the factors of the products of “lower complexity boundary moduli spaces” (so that the neighborhoods of the moduli spaces take a long time to see the compact parts, and, a fortiori, the rates of mixing between the compact parts and neighborhoods of the boundary are not very fast). Formally, for each , one can produce a subset of volume of vectors leading to WP geodesics traveling in the -thin part of (= -neighborhood of the boundary of ) for a time . In other words, there is a subset of volume of geodesics taking time at least to visit the –thick part of (= complement of the -thin part of ). Hence, there are certain non-negligible (volume ) subsets of the -thin part of taking longer and longer (time at least) to mix with the -thick part of as , so that the rate of mixing of the WP flow can not be very fast (i.e., the rate of mixing is not exponential, and, actually, not even a high degree polynomial).
This argument is not hard to formalize once one dispose of adequate estimates on the geometry of the WP metric near the boundary of (for instance, it occupies just 1 page of our note), and we will see it again in this blog in the last post of our series “Dynamics of the Weil-Petersson flow”.
In particular, it is not surprising that we were aware of this argument since 2010: in fact, as far as I can remember, this argument showed up in one of the discussions we had during the intervals of the talks of this conference in honor of Wolpert’s 60th birthday.
On the other hand, this argument breaks down in the case of the moduli spaces of four-times punctured spheres and once-punctured torii because the boundary of these particular moduli spaces consists of a single-point (and, thus, it is impossible to travel “almost-parallel” to the boundary in these particular cases).
This leads us to item b) above: as we announce in our note, the WP flow on the unit cotangent bundles of the particular moduli spaces and is actually rapid (faster than any given polynomial) mixing.
Geometrically, this is intuitively explained by the fact that the WP metric near the boundary of and looks like the metric of the surface of revolution of the profile . In particular, one can check that any WP-geodesic not going straight into the cusp (i.e., hitting the single boundary point of these particular moduli spaces) comes back to the -thick part in time . Since the WP metric has uniformly bounded negative curvature in the -thick part and the geodesic flows on negatively curved surfaces tend to be exponentially mixing, it is reasonable to expect rapid mixing (and maybe even exponential mixing) in these particular cases.
Nevertheless, the implementation of this idea is technically subtle because the curvature of the WP metric near the boundary point of and converges to so that the standard dynamical tools (such as “bounded distortion”) can not be employed directly.
As the reader can imagine, the previous paragraph partly explains why we are taking our time to write down our article.
This being said, the reason why we decided to release this research announcement note is now more or less clear. Indeed, after the 2010 paper of Burns, Masur and Wilkinson on the ergodicity of the WP flow, it is natural to ask about rates of mixing of this flow, and, for instance, several colleagues asked us about this property during a workshop on the WP metric in 2012 at Palo Alto. Here, while the argument for the proof of the result in item a) is extremely simple (and it is written since 2010), we kept (more or less) silent about this project until now only because we thought that the details of item b) could be filled out before the end of 2013 (thus allowing us to upload to ArXiv a more complete paper). Of course, since it is now clear that our initial plan of filling in the details of item b) before Christmas 2013 was too optimistic (on one hand) and it is not reasonable to write a paper just with the simple argument showing item a) (on the other hand), we decided that a research announcement note was the best solution to inform our friends of the results on the rates of mixing of the WP flow that we could get so far.
Closing this short post, let us point out that the end of our note contains a remark that item a) above gives a polynomial upper bound on the rates of mixing of the WP flow but no polynomial lower bound for these rates. Heuristically speaking, we think that one of the difficulties in showing such lower bounds is the absence of precise asymptotic estimates on how the curvatures of the WP metric approach zero (because such estimates would control how close to a product metric is the WP metric near the strata of the boundary of the moduli spaces, and, thus for how long can a geodesic travel almost-parallel to the strata of the boundary of the moduli spaces).