Last time, we reduced the proof of Burns-Masur-Wilkinson theorem on the ergodicity (and mixing) of the Weil-Petersson geodesic flow to a certain estimate for the first derivative of a geodesic flow on negatively curved manifolds (cf. Theorem 11 in this post) and Burns-Masur-Wilkinson ergodicity criterion for geodesic flows on some negatively curved manifolds (cf. Theorem 1 in this post).
The plan for this post is the following. After quickly reviewing in Section 1 below some basic features of the geometry of tangent bundles of Riemannian manifolds, we will estimate the first derivative of geodesic flows on certain negatively curved manifolds in terms its sectional curvatures (as promised last time). Finally, we will complete today’s discussion by proving the first part of Burns-Masur-Wilkinson ergodicity criterion (i.e., we will show that any geodesic flow verifying the assumptions of Burns-Masur-Wilkinson is non-uniformly hyperbolic in the sense of Pesin’s theory), while leaving the second part of Burns-Masur-Wilkinson ergodicity criterion (i.e., the verification of ergodicity via Hopf’s argument) for the next post of this series.
1. Geometry of tangent bundles
1.1. Riemannian metrics, Levi-Civitta connections and Riemannian curvature tensors
Let be a Riemmanian manifold and denote by its Riemannian metric of .
Let be the associated Levi-Civita connection, i.e., the unique connection (“notion of parallel transport”) that is symmetric and compatible with the Riemannian metric . Given a curve on , the covariant derivative along is
(and it should not be confused with ). Sometimes we will also denote the covariant derivative simply by when the curve is implicitly specified: for example, given a vector field along a curve (of footprints), we write where is an extension of to .
In this setting, recall that a curve is a geodesic if and only if for all .
Since the equation is a first order ODE (in the variables ), we have that geodesics are determined by the initial vector . Furthermore, any geodesic has constant speed, i.e., the quantity measuring the square of size (norm) of the tangent vector is constant along : in fact, using the compatibility between and , one gets
for all .
The lack of commutativity of the Levi-Civitta connection is measured by the Riemannian curvature tensor
In terms of the Riemannian curvature tensor , the sectional curvature of a -plane spanned by two vectors and is
1.2. The tangent bundle to a tangent bundle
The tangent bundle of the tangent bundle of is a bundle over in three natural ways:
- (a) where is the natural projection;
- (b) where is the natural projection;
- (c) where is defined as follows: given tangent at to a curve , we set where is the curve of footprints of the vectors ;
In this context, the vertical, resp. horizontal, subbundle of is , resp. . The vertical, resp. horizontal, subbundle is naturally identified with via , resp. . The vertical subbundle is transverse to the horizontal subbundle and the fiber of at can be identified via the map .
Geometrically, the roles of the vertical and horizontal subbundles are easier to understand in the following way. Given an element of tangent to a curve with , let be the curve of footprints of in . In this setting, the identification of with a pair of vectors via the horizontal and vertical subbundles simply amounts to take
In other terms, the component of in the horizontal subbundle measures how fast is moving in while the component of in the vertical subbundle measure how fast is moving in the fibers of .
This way of thinking as a bundle over leads to the following natural Riemannian metric on : given , we define
This metric is called Sasaki metric and the geometry of with respect to this Riemannian metric will be useful in our study of geodesic flows.
Remark 1 Sasaki metric is induced by the symplectic form
in the sense that
where . The symplectic form is the pullback of the canonical symplectic form on the cotangent bundle by the map associating to the linear functional .
For the reader’s convenience, let us mention the following three useful facts about Sasaki metric:
- Sasaki showed that the fibers of the tangent bundle are totally geodesic submanifolds of equipped with Sasaki metric;
- A parallel vector field on viewed as a curve on is a geodesic for Sasaki metric that is always orthogonal to the fibers of ;
- by Topogonov comparison theorem, for close to , one has
where is the vector obtained by parallel transporting along the geodesic connecting to and is the distance associated to Sasaki metric; here, how close must be from depends only on the sectional curvatures of Sasaki metric in a neighborhood of ;
2. First derivative of geodesic flows and Jacobi fields
2.1. Computation of the first derivative of geodesic flows
Let be the geodesic flow associated to a Riemannian manifold . By definition, given a tangent vector , we define where is the unique geodesic of with . Here, it is worth to point out that the geodesic flow is always locally well-defined but it might be globally ill-defined. Moreover, the geodesic flow preserves the Liouville measure (i.e., the volume form on induced by Riemannian metric of ).
We want to describe and, from the definition of first derivative, this amounts to study (-parameter) variations of geodesics.
More precisely, let be a (smooth) map such that, for each , is a geodesic of . Intuitively, is a one-parameter variation of the geodesic .
Define the vector field along the geodesic . It is well-known that satisfies the Jacobi equation
where is the covariant derivative (along ) and is the Riemannian curvature tensor. In other terms, is a Jacobi field, i.e., a vector field satisfying Jacobi’s equation.
Observe that Jacobi’s equation is a second order linear ODE. In particular, a Jacobi field is determined by the initial data .
The pair corresponds to the tangent vector at to the curve in (under the identification described above [in terms vertical and horizontal subbundles]). Indeed, the curve of footprints of is , so that the tangent vector at of is represented by
Here, the symmetry of the Levi-Civitta connection was used.
Similarly, the pair represents the tangent vector at to the curve . Therefore, represents
In summary, Jacobi fields are intimately related to the first derivatives of geodesic flows:
2.2. Perpendicular Jacobi fields and invariant subbundles
A concrete example of Jacobi field along a geodesic is : indeed, in this context, and , so that Jacobi’s equation is trivially verified. Geometrically, this Jacobi field correspond to a trivial variation of the geodesic where the initial point moves along and/or the speed of the parametrization of changes, i.e., .
In general, a Jacobi field along a geodesic that is tangent to has the form for some : in fact, for with , one has , so that Jacobi’s equation reduces to , i.e., for all .
Hence, a Jacobi field along a geodesic is interesting only when it is not completely tangent to the geodesic, or, equivalently, when it has some non-trivial component in the perpendicular direction to the geodesic.
A Jacobi field along a geodesic has the following geometrical properties:
- the component of makes constant angle with , i.e., the quantity is constant;
- if both components of are orthogonal to at some point, then they stay orthogonal all along , i.e., if and for some , then for all ;
We say that a Jacobi field along a geodesic is a perpendicular Jacobi field whenever both components of are orthogonal to .
From the properties of Jacobi fields discussed above, we see that any Jacobi field along a geodesic has a decomposition
where is a perpendicular Jacobi field and is a Jacobi field tangent to .
After this little digression on Jacobi fields, let us use them to introduce relevant invariant subbundles under the first derivative of a geodesic flow .
We begin by recalling that the norm of a tangent vector stays constant along its -orbit, i.e., preserves the energy hypersurfaces (for each ). In particular, the first derivative of the geodesic flow preserves the tangent bundle (to the unit tangent bundle of M).
We affirm that, under the identification for , the fiber (of the subbundle of ) corresponds to the set of pairs with .
In fact, note that an element of is tangent at to a variation of geodesics parametrized by arc-length, i.e.,
for all , such that the geodesic satisfies and the Jacobi field corresponding to verifies .
The desired property now follows from the following calculation:
The invariant subbundle itself admits a decomposition into two invariant subbundles, namely,
where is the vector field generating the geodesic flow and is the orthogonal complement of . In fact, under the identification for , the vector is and the elements of have the form with , . In particular, the -invariance of follows from the fact (mentioned above) that a Jacobi field satisfying and for some is a perpendicular Jacobi field (i.e., and for all ).
In summary, the action of on has two complementary invariant subbundles, namely, the span of the vector field generating the geodesic flow and its orthogonal consisting of perpendicular Jacobi fields. Since acts isometrically in the direction of , our task is reduced to study the action of on perpendicular Jacobi fields.
2.3. Matrix Jacobi and Ricatti equations
We want to describe the matrix of acting on the vector space of perpendicular Jacobi fields. For this sake, let be an orthonormal basis for the tangent space of , and denote by the parallel transport of this orthonormal basis along the geodesic .
Define the matrix whose entries are
where is the Riemannian curvature tensor.
Note that any Jacobi field along can be written as . In this setting, Jacobi equation becomes
and, as usual, a solution is determined by the values and .
We can write solutions of the Jacobi equation above in a practical way by considering a matrix solution of the matrix Jacobi equation:
If is non-singular, the matrix
satisfies the matrix Ricatti equation
for any two columns and of . Here, is the standard symplectic form of .
2.4. An estimate for the first derivative of a geodesic flow
After these preliminaries on the geometry of tangent bundles, geodesic flows and Jacobi fields, we are ready to prove the following result stated as Theorem 11 in our previous post (but whose proof was postponed for this post).
Theorem 2 Let be a negatively curved manifold. Let and consider a geodesic. Suppose that is a Lipschitz function such that, for each , the sectional curvature of any plane containing is greater than or equal to , and denote by the solution of Ricatti’s equation
with initial data . Then, the first derivative of the geodesic flow at time satisfies the estimate
We begin by estimating these quantities for two special subclasses of perpendicular Jacobi fields defined as follows. Let and be the (fundamental) solutions of the matrix Jacobi equation
with initial data and . Note that, by definition, all Jacobi fields with , resp. all Jacobi fields with , have the form , resp. , i.e., they are obtained by applying the matrices , resp. , to a vector , resp. . In this setting, the “other” component , resp. (of the Jacobi field , resp. , viewed as a tangent vector to ) can be recovered by applying the matrix , resp. , to , resp. .
Remark 3 Very roughly speaking, the idea behind the choice of the subclasses and is that are Jacobi fields belonging to a certain “stable cone” and are Jacobi fields belonging to a certain “unstable cone” (compare with the discussion in the next Section).
Our first lemma says that the tangent vectors associated to Jacobi fields as above do not growth in forward time.
Proof: One of the consequences of negative sectional curvatures along is the fact that the functions and are strictly convex for any perpendicular Jacobi field (see, e.g., Eberlein’s survey).
In our context, this implies that is a (strictly) convex function decreasing from to in the interval . Therefore,
Since and for close to (because ), we deduce that
This completes the proof of the lemma.
Our second lemma says that the the growth in forward time of tangent vectors associated to Jacobi fields as above is reasonably controlled in terms of the solution of Ricatti’s equation with (where is the Lipschitz function controlling some sectional curvatures of ).
Proof: By definition, . Thus, and, a fortiori,
On the other hand, since , we see that and, hence,
These inequalities show that the proof of the lemma is complete once we can prove that for all .
In this direction, let us observe that the matrix is symmetric because it verifies (in a trivial way) the condition of Remark 2. Therefore, the norm of is given by the expression
where ranges from all unit vectors. In particular, our task is reduced to show that
for all unit vectors , where .
From the matrix Ricatti equation, we see that
Since the Lipschitz function controls the sectional curvatures (of planes containing ) along and the matrix is symmetric, we can estimate the right-hand side of the previous inequality as
On the other hand, since is a unit vector, the Cauchy-Schwarz inequality implies that . Therefore, the right-hand side of the previous inequality is bounded by
From this differential inequality and the facts that and , we can easily deduce that for all from the standard continuity argument.
Finally, we can complete the proof of the lemma by observing that the symmetric matrix is positive definite for : this follows from the facts that and satisfies matrix Ricatti equation associated to a negatively curved manifold (cf. Eberlein’s book). Therefore, and for all and all unit vector , so that
Once we know how to control the growth of and for Jacobi fields and as above, the idea to estimate the growth of for an arbitrary perpendicular Jacobi field (thus completing the proof of Theorem 2) is to produce a decomposition of the form
where and the norms of and are controlled in terms of the norm of .
For this sake, define
and we set and .
First, note that the vector is well-defined, i.e., the matrix is invertible. Indeed, we already saw that the matrices and are symmetric (because they satisfy (in a trivial way) the condition of Remark 2) and that the matrix is positive definite (because satisfies matrix Ricatti equation, the manifold is negatively curved and imply that is positive-definite for (cf. Eberlein’s book). Furthermore, all eigenvalues of the matrix are : in fact, any eigenvalue of has the form for some unit vector , and because is a convex function (see, e.g., Eberlein’s survey) decreasing from to in the interval (with ). Therefore, the matrix is a symmetric matrix whose eigenvalues are and, hence, is an invertible matrix satisfying
Secondly, we claim that the Jacobi fields and give the desired decomposition. In fact, since , and are Jacobi fields, our claim follows from the facts that and .
Finally, let us estimate the (Sasaki) norms of and in terms of . We begin by observing that it suffices to estimate the Sasaki norm of because
On the other hand, the (Sasaki) norm of is not difficult to bound:
Since (cf. the proof of Lemma 4) and , we can estimate the right-hand side of the previous inequality by
This completes the proof of Theorem 2.
3. Hyperbolicity of geodesic flows on certain negatively curved manifolds
In this section, we will partly fulfill our promise in our previous post by giving the first steps towards the proof of Burns-Masur-Wilkinson ergodicity criterion:
Theorem 5 (Burns-Masur-Wilkinson) Let be the quotient of a contractible, negatively curved, possibly incomplete, Riemannian manifold by a subgroup of isometries of acting freely and properly discontinuously. Denote by the metric completion of and the boundary of .Suppose that:
- (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).
More precisely, our plan for the rest of this post is to show the non-uniform hyperbolicity of the geodesic flow described in the statement above. Then, we will leave the proof of the ergodicity of (via Hopf’s argument) for the next post of this series.
We start by noticing that has finite -volume: this is an easy consequence of the compactness of (assumption (II)) and the volumetrically cusp-like assumption (III) on .
Next, let us check that the geodesic flow in the statement of Burns-Masur-Wilkinson ergodicity criterion is defined for all time for almost every initial data . For this sake, denote by the natural projection and set
and, a fortiori,
In particular, since the geodesic flow preserves the measure , our task of showing that is defined for all time for almost every initial data is reduced to prove that has zero -measure.
In order to compute the -measure of , let us estimate the -measure of for each along the following lines. Note that
where consists into the unit tangent vectors flowing into for some time between and . By definition, , so that
where . Here, we used the fact that is -invariant (for the first inequality) and the assumption (III) (for the second inequality). It follows that
for all . Hence, has zero -measure and is defined for all time for almost all initial data.
Remark 4 The reader certainly noticed that we do not the full strength of assumption (III) to deduce the long-term existence of at almost every point: in fact, the weaker condition works equally well. Nevertheless, we will see below that the full strength of assumption (III) is helpful to ensure the existence of Lyapunov exponents for the geodesic flow .
Now, let us show that the geodesic flow is non-uniformly hyperbolic in the sense of Pesin theory, i.e., all (transverse) Lyapunov exponents are non-zero.
We start by verifying that the Lyapunov exponents of are well-defined (at almost every point): by Oseledets multiplicative ergodic theorem, it suffices to check the -integrability of the derivative cocycles and associated to the time- and time-maps and , that is,
By symmetry (or reversibility of the geodesic flow), we have to consider only the -integrability of . We estimate the integral above for by noticing that
Since is compact (by assumption (II)), we need to show only that the series above is convergent and this is not hard to see: on one hand, we already saw that for some (as a consequence of assumption (III), and, on the other hand, on by assumption (VI), so that
By Oseledets theorem, once we know the -integrability of the derivative cocycle, we have that, for almost every , there are real numbers
called Lyapunov exponents and a -invariant splitting
into Lyapunov subspaces such that, for every ,
In the context of a geodesic flow , recall that the derivative cocycle preserves the decomposition , and acts isometrically along and preserves . This implies that the Lyapunov exponent of along is zero and the derivative cocycle has Lyapunov exponents counted with multiplicity (i.e., we count -times the Lyapunov exponent ) along .
Remark 5 In fact, the derivative cocycle preserves a natural symplectic form on . In particular, the Lyapunov exponents are organized in a symmetric way around the origin in the sense that is a Lyapunov exponent whenever is a Lyapunov exponent.
By definition, is called non-uniformly hyperbolic whenever all Lyapunov exponents along (sometimes called transverse Lyapunov exponents) are non-zero.
In our context (of the statement of Burns-Masur-Wilkinson ergodicity criterion), we will prove the non-uniform hyperbolicity of by exploiting the negative curvature of . More concretely, the negative curvature of implies that:
- for any non-trivial perpendicular Jacobi field , the functions and are strictly convex (thanks to Jacobi’s equation);
- for each geodesic ray and for each , there exists an unique perpendicular Jacobi field along with such that
for all .
See, e.g., Eberlein’s book for more explanations. In the literature, is called an unstable Jacobi field and it is usually constructed as the limit where is the Jacobi field with and . Similarly, we can define stable Jacobi fields along geodesic rays by reversing the time of the geodesic flow. The Figure 2 above illustrates stable (“blue”) and unstable (“red”) Jacobi fields along a vertical geodesic in the hyperbolic plane.
We will discuss stable and unstable Jacobi fields in more details in the next post of this series (because they describe the stable and unstable manifolds of and Hopf’s argument depend crucially on the features of stable and unstable manifolds). For now, we just need to know that, if is negatively curved and is defined for all time at , then
In other terms, where and are -dimensional subspaces related to stable and unstable Jacobi fields. See, e.g., Eberlein’s book for a proof of this fact.
In this setting, the non-uniform hyperbolicity of is a direct consequence of the following lemma relating stable and unstable Jacobi fields to Lyapunov subspaces:
Lemma 6 There exists a -invariant subset of full -measure such that
Proof: Denote by the set of unit vectors such that:
- is defined for all time ;
- the Lyapunov exponents and Lyapunov subspaces are defined for ;
- is uniformly recurrent under in the sense that, for any neighborhood of , there exists such that the sets have Lebesgue measure for all sufficiently large.
Note that is -invariant and it has full -measure: our previous discussion showed that the first two conditions hold for almost every and the third condition holds in a full measure subset thanks to Birkhoff’s ergodic theorem.
We affirm that satisfies the conclusions of the lemma. In fact, by the reversibility of the geodesic flow , it suffices to show that
for all .
For this sake, given , we fix a neighborhood of and a real number such that if is an unstable Jacobi field along a geodesic with , then
The choice of and is possible because is negatively curved and is an increasing strictly convex function whose second derivative is controlled by Jacobi’s equation.
Since is uniformly recurrent for , we have that
for all . Because , we know that has Lebesgue measure for some and for all sufficiently large. Therefore, for any unstable Jacobi field along , one has
for all sufficiently large. It follows from the definitions that
for any , and, hence,
Similarly, . Because , these inclusions must be equalities and the proof of the lemma is complete.
For later reference, we summarize the results of this section in the following statement:
Theorem 7 Under the assumptions (I) to (VI) in Theorem 5 above, the geodesic flow is non-uniformly hyperbolic: more concretely, there exists a subset of full -measure such that the -invariant splitting
into the flow direction and the spaces and of stable and unstable Jacobi fields along have the property that
for all and .