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.
1. Ergodicity of WP flow: outline of proof
The initial idea to prove Burns-Masur-Wlkinson theorem is the “usual” argument for the proof of ergodicity of a system exhibiting some hyperbolicity, namely, Hopf’s argument.
The general scheme of this argument is the following. Given a smooth flow on a compact Riemannian manifold preserving the corresponding volume measure and a continuous observable , we consider the future and past Birkhoff averages:
By Birkhoff’s ergodic theorem, for -almost every , the quantities and exist and, actually, they coincide . In the literature, a point such that , exist and is called a Birkhoff generic point (with respect to ).
By definition, the ergodicity of (with respect to ) is equivalent to the fact that the functions and are constant at -almost every point.
In order to show the ergodicity of a flow with some hyperbolicity, Hopf observes that the function , resp. , is constant along stable, resp. unstable, sets
i.e., whenever , resp. whenever . We leave the verification of this fact as an exercise to the reader.
In the case of a volume-preserving Anosov flow (sometimes called uniformly hyperbolic flow) on , we know that the stable and unstable sets are immersed submanifolds. Moreover, if one forgets about the flow direction, the stable and unstable manifolds have complementary dimensions and intersect transversely. Hence, given two points (lying in distinct orbits of ), we can connect them using pieces of stable and unstable manifolds as shown in the figure below:
In particular, this indicates that a volume-preserving Anosov flow is ergodic because the functions and are constant along stable and unstable manifolds, they coincide almost everywhere and any pair of points can be connected via pieces of stable and unstable manifolds. However, this argument towards ergodicity of is not complete yet: indeed, one needs to know that the intersection points between the pieces of stable and unstable manifolds connecting and are Birkhoff generic in order to conlude that .
In the original context of his article, Hopf studies a geodesic flow of a compact surface of constant negative curvature, and he uses the fact that the stable and unstable manifolds form foliations by curves to deduce that the intersection points can be taken to be Birkhoff generic points. Indeed, since the stable and unstable foliations are in his context, Hopf applies Fubini’s theorem to the set of full -volume consisting of Birkhoff generic points in order to ensure that almost all stable and unstable manifolds/curves and intersect in a subset of total length measure of and .
On the other hand, it is known that the stable and unstable manifolds of a general Anosov flow (such as geodesic flows on compact manifolds of variable negative curvature) do not form necessarily a -foliation, but only Hölder continuous foliations (see e.g. the papers of Anosov and/or Hasselblatt for concrete examples). In particular, this is an obstacle to the argument à la Fubini of the previous paragraph. Nevertheless, Anosov showed that the stable and unstable foliations of a smooth Anosov flow are always absolutely continuous, so that one can still apply Fubini’s theorem to conclude ergodicity along the lines of Hopf’s argument presented.
In summary, we know that a smooth () volume-preserving Anosov flow on a compact manifold is ergodic thanks to Hopf’s argument and the absolute continuity of stable and unstable foliations.
Remark 1 Robinson-Young showed that the stable and unstable foliations of a Anosov system are not necessarily absolutely continuous. In particular, the smoothness () assumption on the Anosov flow is necessary for the ergodicity argument described above.
Remark 2 The absolute continuity of a foliation invariant under some system depends on some hyperbolicity. In fact, Shub-Wilkinson constructed examples of invariant foliations of certain partially hyperbolic diffeomorphisms failing to satisfy Fubini’s theorem in the sense that each leaf of this foliation (along which the dynamics is central/neutral) intersects a set of full volume exactly at one point! This phenomenon is sometimes referred to as Fubini’s nightmare in the literature (see, e.g., this article of Milnor) and sometimes a foliation “failing” Fubini’s theorem is called a pathological foliation (see, e.g., these pictures by Wilkinson for some examples of such foliations)
After this brief sketch of Hopf’s argument for smooth volume-preserving Anosov flows on compact manifolds, let us explain the difficulties of extending this argument to the setting of WP flow.
As we already mentioned (cf. item (d) of the table above), the WP flow is singular hyperbolic. In a nutshell, this means that, even though WP flow is not uniformly hyperbolic (Anosov), it is non-uniformly hyperbolic in the sense of Pesin theory and it satisfies some hyperbolicity estimates along pieces of orbits staying in compact parts of moduli space.
In particular, thanks to Pesin’s stable manifold theorem, the stable and unstable sets of almost every point are immersed submanifolds, and, if we forget about the flow direction, the stable and unstable manifolds have complementary dimensions. Furthermore, the stable and unstable manifolds are part of absolutely continuous laminations. Here, it is important that the dynamics is sufficiently smooth (see, e.g., this paper of Pugh, and this preprint of Bonatti, Crovisier and Shinohara).
Thus, this gives hopes that Hopf’s argument could be applied to show the ergodicity of volume-preserving non-uniformly hyperbolic systems.
However, by inspecting the figure 1 above, we see that Hopf’s argument relies on the fact that stable and unstable manifolds of Anosov flows have a nice, well-controlled, geometry.
For instance, if we start with a point and we want to connect it with pieces of stable and unstable manifolds to a point at a large distance, we have to make sure that the pieces of stable and unstable manifolds used in figure 1 are “uniform”, e.g., they are graphs of definite size and bounded curvature with respect to the splitting into stable and unstable directions, and, moreover, the angles between the stable and unstable directions are uniformly bounded away from zero.
Indeed, if the pieces of stable and unstable manifolds get shorter and shorter, and/or if they “curve” a lot, and/or the angles between stable and unstable directions are not bounded away from zero, one might not be able to reach/access from with stable and unstable manifolds:
As it turns out, while these kinds of non-uniformity do not occur for Anosov flows, they can actually occur for certain non-uniformly hyperbolic systems. More precisely, the sizes and curvatures of stable and unstable manifolds, and the angles between stable and unstable directions of a general non-uniformly hyperbolic system vary only measurably from point to point.
In particular, this excludes a priori a naive generalization of Hopf’s ergodicity argument for non-uniformly hyperbolic systems, and, in fact, there are concrete examples by Dolgopyat, Hu and Pesin of volume-preserving non-uniformly hyperbolic systems with countably many ergodic components consisting of invariant sets of positive volumes that are essentially open (and, as a matter of fact, this example is “sharp” in the sense that Pugh and Shub showed that a volume-preserving non-uniformly hyperbolic system has at most countably many ergodic components).
In summary, the ergodicity of a non-uniformly hyperbolic system depends on the particular dynamical features of the given system.
In this direction, there is a vast literature dedicated to the construction of large classes of ergodic non-uniformly hyperbolic systems: for example, the ergodicity of several classes of billiards was shown by Sinai, Bunimovich, Chernov-Bunimovich-Sinai among others (see also the book of Chernov-Markarian) and the ergodicity of non-uniformly hyperbolic systems exhibiting some partial hyperbolicity (or some “dominated splitting”) was shown by Pugh-Shub, Rodriguez-Hertz, Tahzibi, Burns-Wilkinson, Rodriguez-Hertz– Rodriguez-Hertz–Ures among others.
For the proof of their ergodicity result for the WP flow, Burns, Masur and Wilkinson take part of their inspiration from the work of Katok-Strelcyn where Pesin’s theory (of existence and absolute continuity of stable and unstable manifolds) is extended to singular hyperbolic systems.
In a nutshell, the basic philosophy behind Katok-Strelcyn’s work is the following. Given a non-uniformly hyperbolic system with some non-trivial singular set, all dynamical features predicted by Pesin theory in virtue of the (non-uniform) exponential contraction and expansion are not affected if the loss of control on the system is at most polynomial as one approaches the singular set. In other terms, the hyperbolic (exponential) behavior of a singular system is not disturbed by the presence of a singular set where the first two derivatives of the system lose control in a polynomial way. In particular, this hints that Hopf’s argument can be extended to singular hyperbolic systems with polynomially bad singular sets.
In this direction, Burns-Masur-Wilkinson shows the following ergodicity criterion for singular hyperbolic geodesic flows (cf. Theorem 3.1 of Burns-Masur-Wilkinson’s paper).
Let be the quotient of a contractible, negatively curved, possibly incomplete, Riemannian manifold by a subgroup of isometries of acting freely and properly discontinuously.
By slightly abusing notation, we denote by the metrics on and induced by the Riemannian metric of .
We consider the metric completion of the metric space , i.e., the (complete) metric space consisting of all equivalence classes of Cauchy sequences under the relation if and only if equipped with the metric , and we define the boundary .
- (I) the universal cover of is geodesically convex, i.e., for every , there exists an unique geodesic segment in connecting and .
- (II) the metric completion of is compact.
- (III) the boundary is volumetrically cusplike, i.e., for some constants and , the volume of a -neighborhood of the boundary satisfies
for every .
- (IV) has polynomially controlled curvature, i.e., there are constants and such that the curvature tensor of and its first two derivatives satisfy the following polynomial bound
for every .
- (V) has polynomially controlled injectivity radius, i.e., there are constants and such that
for every (where denotes the injectivity radius at ).
- (VI) The first derivative of the geodesic flow is polynomially controlled, i.e., there are constants and such that, for every infinite geodesic on and every :
Then, the Liouville (volume) measure of is finite, the geodesic flow on the unit cotangent bundle of is defined at -almost every point for all time , and the geodesic flow is non-uniformly hyperbolic (in the sense of Pesin’s theory) and ergodic.
Actually, the geodesic flow is Bernoulli and, furthermore, its metric entropy is positive, finite and is given by Pesin’s entropy formula (i.e., it is equal to the sum of positive Lyapunov exponents of counted with multiplicities).
Once Burns, Masur and Wilkinson have Theorem 3 in their toolbox, they prove the ergodicity result for WP flow in Theorem 1 by checking that the moduli space of curves (Riemann surfaces) equipped with the WP metric satisfies the six items (I) to (VI) above. Here, it is worth to point out that the most delicate items are (IV) and (VI), and Burns, Masur and Wilkinson verify them for the WP metric with the help of important previous works of McMullen and Wolpert (see these papers here).
In any case, this completes the outline of the proof of Burns-Masur-Wilkinson theorem on the ergodicity of WP flow.
2. Rates of mixing of WP flow
As we mentioned above, both Teichmüller and WP flows are uniformly hyperbolic in compact parts of the moduli space of curves. Since a uniformly hyperbolic system is exponentially mixing, the sole thing preventing an exponential rate of mixing for these flows is the possibility that a “big” set of orbits spends a “lot” of time near infinity (or rather the boundary of the moduli space) before coming back to the compact parts.
In the case of Teichmüller flow, the volume in Teichmüller metric of a -neighborhood of the boundary of moduli space is exponentially small (of order where denotes any fixed positive real number strictly smaller than ; cf. Corollary 2.16 of Avila-Gouëzel-Yoccoz paper).
Intuitively, this says that the “probability” that an orbit spends a long time near the boundary of moduli space is exponentially small (cf. Theorem 2.15 of Avila-Gouëzel-Yoccoz paper). In particular, the excursions near infinity of most orbits is not long enough to disrupt the exponential rate of mixing “imposed” by hyperbolic dynamics of the Teichmüller flow on compact parts. Of course, this is merely a vague intuition behind the exponential mixing of the Techmüller flow and the curious reader is encouraged to consult the articles of Avila-Gouëzel-Yoccoz and and Avila-Gouëzel for detailed explanations.
On the other hand, in the context of the WP flow, we will see that the volume in WP metric of -neighborhood of the boundary of moduli space is (see Lemma 6.1 of Burns-Masur-Wilkinson paper).
Therefore, the “probability” that an orbit of WP flow spends a long time near infinity might be only polynomially small but not exponentially small. In particular, this possibility might conspire against an exponential mixing of WP flow.
In fact, in our joint work with Burns, Masur and Wilkinson, we construct a subset of volume of orbits of WP flow staying near infinity for a time (at least). For this sake, we use some estimates of Wolpert (see also Propositions 4.11, 4.12 and 4.13 in Burns-Masur-Wilkinson paper) saying that the geometry of WP metric on the moduli space of curves of genus looks like a product of the WP metrics on the moduli spaces of curves of lower genera . In particular, the set is chosen to correspond to geodesics travelling almost parallel to one of the factors of the product for a relatively long time.
On the other hand, the existence of such sets means that the rate of mixing of WP flow can not be very fast.
Indeed, by taking a “smooth approximation” of the characteristic function of (i.e., supported on and ), and by letting be a fixed smooth function supported on the compact part (away from infinity), we see that
for . In fact, the second equality follows because is supported in the compact part of the moduli space, is supported on and the set is disjoint from the compact part for (by construction of ), so that for . Therefore, at time , we deduce that , and, hence, the correlation functions associated to WP flow can not decay faster than a polynomial function of degree of as the time .
Finally, let us remark that this argument does not work in genus because the crucial fact (in the construction of the set ) that the WP metric looks like the product of WP metrics in moduli spaces of lower genera breaks down in genus : indeed, in this situation, the moduli space is naturally compactified by adding a single point (because the moduli space in lower genus is trivial) and so the WP metric does not behave like a product (or, more precisely, no sectional curvature approaches zero as we get close to infinity). In this case, Burns, Masur, Wilkinson and myself shows that this “absence of zero curvatures at infinity” actually implies that the rate of mixing of the WP flow on the moduli space of torii is rapid, i.e., faster than any polynomial function of .
Concluding this section (and today’s post), let us observe that we do not know if the rate of mixing of the WP flow on moduli space of curves of genus is genuinely polynomial.
Indeed, recall that the naive intuition says that the rate of mixing is polynomial if we can show that most orbits do not spend long time near infinity.
Of course, this would not be the case if the WP metric is very close to a product metric, or, more precisely, if some sectional curvatures of WP metric are very close to zero: in fact, the structure of a product metric near infinity would allow for several orbits to travel almost parallel to the factors of the product (and, hence, near infinity) for a very long time.
So, we need estimates saying how fast the sectional curvatures of WP metric approach zero as one gets close to infinity, and, unfortunately, the best estimates available so far (due to Wolpert) do not give this type of information (because of certain potential cancellations in Wolpert’s calculations).