For the last installment of this series, our goal is to discuss the rates of mixing of the Weil-Petersson (WP) geodesic flow on the unit tangent bundle of the moduli space of Riemann surfaces of genus with punctures for .
However, before entering into the mathematical discussion strictly speaking, let me take the opportunity to dedicate this blog post to the memory of two Russian mathematicians who passed away earlier this month: Dmitri Anosov and Nikolai Chernov. Among their several well-known contributions in Dynamical Systems, we can quote:
Of course, the list of contributions of Anosov and Chernov to Dynamical Systems is vast: each of them wrote more than 90 research articles and books about the features of systems with some hyperbolicity (such as geodesic flows on negatively curved manifolds and chaotic billiards) among other topics.
In particular, it is out of the scope of this post to provide detailed descriptions of the works of these two very influential dynamicists.
On the other hand, as a form of “small compensation”, let me say that the second section of this post (about rates of the WP flow on the modular surface) briefly discusses some of the ideas advanced by these two mathematicians.
Concerning the rates of mixing of the WP flow, let us recall that, by Burns-Masur-Wilkinson theorem (cf. Theorem 1 in the first post of this series), the WP flow on is mixing with respect to the Liouville measure whenever .
By definition of the mixing property, this means that the correlation function converges to as for any given -integrable observables and . (See, e.g., the section “ formulation” in this Wikipedia article about the mixing property.)
Given this scenario, it is natural to ask how fast the correlation function converges to zero. In general, the correlation function can decay to (as a function of ) in a very slow way depending on the choice of the observables (see, e.g., this blog post of Climenhaga for some concrete examples). Nevertheless, it is often the case (for mixing flows with some hyperbolicity) that the correlation function decays to with a definite (e.g., polynomial, exponential, etc.) speed when restricting the observables to appropriate spaces of “reasonably smooth” functions.
In other words, given a mixing flow (with some hyperbolicity), it is usually possible to choose appropriate functional (e.g., Hölder, , Sobolev, etc.) spaces and such that
- for some constants , and for all (polynomial decay),
- or for some constants , and for all (exponential decay).
Evidently, the “precise” rate of mixing of the flow (i.e., the sharp values of the constants , and/or above) depend on the choice of the functional spaces and (as they might change if we replace observables by observables say). On the other hand, the qualitative speed of decay of , that is, the fact that decays polynomially or exponentially as whenever and are “reasonably smooth”, remains unchanged if we select and from a well-behaved scale of functional (like spaces, , or spaces, ). In particular, this partly explains why in the Dynamical Systems literature one simply says that a given mixing flow has “polynomial decay” or “exponential decay”: usually we are interested in the qualitative behavior of the correlation function for reasonably smooth observables, but the particular choice of functional spaces and is normally treated as a “technical detail”.
After this brief description of the notion of rate of mixing (speed of decay of correlation functions), we are ready to state the main result of this post.
Theorem 1 (Burns-Masur-M.-Wilkinson) The rate of mixing of the WP flow on is:
- at most polynomial when ;
- rapid (faster than any polynomial) when .
Remark 1 This result was announced as Theorem 2 in the first post of this series and also in this preprint here. Since then, Burns, Masur, Wilkinson and myself found some evidence indicating that the Weil-Petersson geodesic flow on is actually exponentially mixing when . The details will hopefully appear in the forthcoming paper (currently still in preparation).
Remark 2 An open problem left by Theorem 1 is to determine the rate of mixing of the WP flow on for . Indeed, while this theorem provides a polynomial upper bound for the rate of mixing in this setting, it does not rule out the possibility that the actual rate of mixing of the WP flow is sub-polynomial (even for reasonably smooth observables). Heuristically speaking, we believe that the sectional curvatures of the WP metric control the time spend by WP geodesics near the boundary of . In particular, it seems that the problem of determining the rate of mixing of the WP flow (when ) is somewhat related to the issue of finding suitable (polynomial?) bounds for how close to zero the sectional curvatures of the WP metric can be (in terms of the distance to the boundary of ). Unfortunately, the best available bounds for the sectional curvatures of the WP metric (due to Wolpert) do not rule out the possibility that some of these quantities get extremely close to zero (see Remark 4 of this post here).
The difference in the rates of mixing of the WP flow on when or in Theorem 1 reflects the following simple (yet important) feature of the WP metric near the boundary of the Deligne-Mumford compactification of .
In the case , e.g., , the moduli space equipped with the WP metric looks like the surface of revolution of the profile near the cusp at infinity (see Remark 6 of this post here). In particular, even though a -neighborhood of the cusp is “polynomially large” (with area ), the Gaussian curvature approaches only near the cusp and, as it turns out, this strong negative curvature near the cusp makes that all geodesic not pointing directly towards the cusp actually come back to the compact part in bounded (say ) time. In other words, the excursions of infinite WP geodesics on near the cusp are so quick that the WP flow on is “close” to a classical Anosov geodesic flow on negatively curved compact surface. In particular, it is not entirely surprising that the WP flow on is rapid.
On the other hand, in the case , the WP metric on has some sectional curvatures close to zero near the boundary of the Deligne-Mumford compactification of (see Theorem 3 and Remark 5 of this post here). By exploiting this feature of the WP metric on for (that has no counterpart for or ), we will build a non-neglegible set of WP geodesics spending a long time near the boundary of before eventually getting into the compact part. In this way, we will deduce that the WP flow on takes a fair (polynomial) amount of time to mix certain parts of the boundary of with fixed compact subsets of .
In the remainder of this post, we will give some details of the proof of Theorem 1. In the next section, we give a fairly complete proof (assuming the results in this previous post, of course) of the polynomial upper bound on the rate of mixing of the WP flow on when . After that, in the final section, we provide a sketch of the proof of the rapid mixing property of the WP flow on . In fact, we decided (for pedagogical reasons) to explain some key points of the rapid mixing property only in the toy model case of a negatively curved surface with one cusp corresponding exactly to a surface of revolution of a profile , . In this way, since the WP metric near the cusp of can be thought as a “perturbation” of the surface of revolution of (thanks to Wolpert’s asymptotic formulas), the reader hopefully will get a flavor of the main ideas behind the proof of rapid mixing of the WP flow on without getting into the (somewhat boring) technical details needed to check that the arguments used in the toy model case are “sufficiently robust” so that they can be “carried over” to the “perturbative setting” of the WP flow on .