Posted by: matheuscmss | May 12, 2014

## Dynamics of the Weil-Petersson flow: proof of Burns-Masur-Wilkinson ergodicity criterion I

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 ${M}$ be a Riemmanian manifold and denote by ${\langle .,. \rangle}$ its Riemannian metric of ${M}$.

Let ${\nabla}$ be the associated Levi-Civita connection, i.e., the unique connection (“notion of parallel transport”) that is symmetric and compatible with the Riemannian metric ${\langle .,. \rangle}$. Given a curve ${c: t\mapsto c(t)}$ on ${M}$, the covariant derivative along ${c}$ is

$\displaystyle D_c:=\nabla_{\dot{c}(t)}=:\frac{D}{dt}$

(and it should not be confused with ${\dot{c}(s)=\frac{d c}{dt}(s)}$). Sometimes we will also denote the covariant derivative simply by ${'}$ when the curve ${c}$ is implicitly specified: for example, given a vector field ${V(t)}$ along a curve ${c}$ (of footprints), we write ${V'(t)=D_c\widehat{V}}$ where ${\widehat{V}}$ is an extension of ${V}$ to ${M}$.

In this setting, recall that a curve ${c}$ is a geodesic if and only if ${D_c \dot{c}(t)=0}$ for all ${t}$.

Since the equation ${D_c \dot{c}(t)=0}$ is a first order ODE (in the variables ${(c,\dot{c})}$), we have that geodesics are determined by the initial vector ${(c(0), \dot{c}(0))}$. Furthermore, any geodesic has constant speed, i.e., the quantity ${\langle \dot{c}(t), \dot{c}(t)\rangle}$ measuring the square of size (norm) of the tangent vector ${\dot{c}(t)}$ is constant along ${c}$: in fact, using the compatibility between ${\nabla}$ and ${\langle.,.\rangle}$, one gets

$\displaystyle \frac{d}{dt}\langle \dot{c}(t), \dot{c}(t)\rangle = 2\langle D_c \dot{c}(t), \dot{c}(t)\rangle = 0$

for all ${t}$.

The lack of commutativity of the Levi-Civitta connection is measured by the Riemannian curvature tensor

$\displaystyle R(A,B)C:=\nabla_A \nabla_B C - \nabla_B \nabla_A C - \nabla_{[A,B]}C.$

In terms of the Riemannian curvature tensor ${R}$, the sectional curvature ${K(A,B)}$ of a ${2}$-plane spanned by two vectors ${A}$ and ${B}$ is

$\displaystyle K(A,B):= \frac{\langle R(A,B)B, A\rangle}{\|A\wedge B\|^2}$

1.2. The tangent bundle to a tangent bundle

The tangent bundle ${TTM}$ of the tangent bundle ${TM}$ of ${M}$ is a bundle over ${M}$ in three natural ways:

• (a) ${TTM\stackrel{D\pi_M}{\rightarrow} TM \stackrel{\pi_M}{\rightarrow} M}$ where ${\pi_M:TM\rightarrow M}$ is the natural projection;
• (b) ${TTM\stackrel{\pi_{TM}}{\rightarrow} TM \stackrel{\pi_M}{\rightarrow} M}$ where ${\pi_{TM}:TTM\rightarrow TM}$ is the natural projection;
• (c) ${TTM\stackrel{\kappa}{\rightarrow} TM \stackrel{\pi_M}{\rightarrow} M}$ where ${\kappa}$ is defined as follows: given ${\xi\in TTM}$ tangent at ${t=0}$ to a curve ${t\mapsto V(t)\in TM}$, we set ${\kappa(\xi):= D_c V(0)}$ where ${c(t)=\pi_M(V(t))}$ is the curve of footprints of the vectors ${V(t)}$;

In this context, the vertical, resp. horizontal, subbundle of ${TTM}$ is ${\textrm{ker}(D\pi_M)}$, resp. ${\textrm{ker}(\kappa)}$. The vertical, resp. horizontal, subbundle is naturally identified with ${TM}$ via ${\kappa}$, resp. ${D\pi_M}$. The vertical subbundle is transverse to the horizontal subbundle and the fiber ${T_vTM}$ of ${TTM}$ at ${v\in T_pM}$ can be identified ${T_pM\times T_pM}$ via the map ${D\pi_M\times\kappa:TTM\rightarrow TM\times TM}$.

Geometrically, the roles of the vertical and horizontal subbundles are easier to understand in the following way. Given an element of ${\xi\in T_vTM}$ tangent to a curve ${V:t\mapsto V(t)\in TM}$ with ${V(0)=v}$, let ${c(t)=\pi_M(V(t))}$ be the curve of footprints of ${V(t)}$ in ${M}$. In this setting, the identification of ${\xi}$ with a pair of vectors ${(v_1, v_2)\in T_pM\times T_pM}$ via the horizontal and vertical subbundles simply amounts to take

$\displaystyle \xi=(v_1,v_2)=(\dot{c}(0), D_cV(0))$

In other terms, the component ${v_1=\dot{c}(0)}$ of ${\xi}$ in the horizontal subbundle measures how fast ${V(t)}$ is moving in ${M}$ while the component ${v_2=D_cV(0)}$ of ${\xi}$ in the vertical subbundle measure how fast ${V(t)}$ is moving in the fibers of ${TM}$.

This way of thinking ${TTM}$ as a bundle over ${M}$ leads to the following natural Riemannian metric on ${TM}$: given ${(v_1,v_2), (w_1,w_2)\in T_vTM}$, we define

$\displaystyle \langle(v_1,v_2), (w_1,w_2)\rangle_{Sas}:= \langle v_1, w_1 \rangle + \langle v_2, w_2 \rangle$

This metric is called Sasaki metric and the geometry of ${TM}$ 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

$\displaystyle \omega((v_1,v_2), (w_1, w_2)):= \langle v_1, w_2\rangle - \langle v_2, w_1\rangle$

in the sense that

$\displaystyle \langle(v_1,v_2), (w_1,w_2)\rangle_{Sas} = \omega((v_1, v_2), J(w_1, w_2))$

where ${J(w_1, w_2) := (-w_2, w_1)}$. The symplectic form ${\omega}$ is the pullback of the canonical symplectic form on the cotangent bundle ${T^*M}$ by the map ${TM\rightarrow T^*M}$ associating to ${v\in TM}$ the linear functional ${\langle v,.\rangle\in T^*M}$.

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 ${TM}$ are totally geodesic submanifolds of ${TM}$ equipped with Sasaki metric;
• A parallel vector field on ${M}$ viewed as a curve on ${TM}$ is a geodesic for Sasaki metric that is always orthogonal to the fibers of ${TM}$;
• by Topogonov comparison theorem, for ${v'\in T_{p'}M}$ close to ${v\in T_pM}$, one has

$\displaystyle \frac{1}{2}(d(p,p')+\|w-v'\|)\leq d_{Sas}(v,v')\leq 2(d(p,p')+\|w-v'\|)$

where ${w\in T_{p'}M}$ is the vector obtained by parallel transporting ${v}$ along the geodesic connecting ${p}$ to ${p'}$ and ${d_{Sas}}$ is the distance associated to Sasaki metric; here, how close ${v'}$ must be from ${v}$ depends only on the sectional curvatures of Sasaki metric in a neighborhood of ${v}$;

2. First derivative of geodesic flows and Jacobi fields

2.1. Computation of the first derivative of geodesic flows

Let ${\varphi_t}$ be the geodesic flow associated to a Riemannian manifold ${M}$. By definition, given a tangent vector ${v\in TM}$, we define ${\varphi_t(v):=\dot{\gamma}_v(t)}$ where ${\gamma_v(s)}$ is the unique geodesic of ${M}$ with ${\dot{\gamma}_v(0)=v}$. 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 ${\varphi_t}$ preserves the Liouville measure (i.e., the volume form on ${TM}$ induced by Riemannian metric of ${M}$).

We want to describe ${D\varphi_t}$ and, from the definition of first derivative, this amounts to study (${1}$-parameter) variations of geodesics.

More precisely, let ${\alpha: (-\varepsilon,\varepsilon)\times (-\varepsilon,\varepsilon)\rightarrow M}$ be a (smooth) map such that, for each ${s}$, ${\alpha(s,.)}$ is a geodesic of ${M}$. Intuitively, ${\alpha}$ is a one-parameter variation of the geodesic ${\gamma(t):=\alpha(0,t)}$.

Define the vector field ${J(t):=\frac{\partial\alpha}{\partial s}(0,t)}$ along the geodesic ${\gamma(t)=\alpha(0,t)}$. It is well-known that ${J}$ satisfies the Jacobi equation

$\displaystyle J''+R(J,\dot{\gamma})\dot{\gamma}=0$

where ${'}$ is the covariant derivative (along ${\gamma}$) and ${R}$ is the Riemannian curvature tensor. In other terms, ${\frac{\partial\alpha}{\partial s}(0,t)}$ is a Jacobi field, i.e., a vector field satisfying Jacobi’s equation.

Fig1. A Jacobi field associated to a variation of geodesics connecting the north and south poles of a round sphere.

Fig 2. Jacobi fields associated to two (blue and red) variations of the geodesic in hyperbolic plane “connecting” ${0}$ to ${\infty}$.

Observe that Jacobi’s equation is a second order linear ODE. In particular, a Jacobi field ${J}$ is determined by the initial data ${(J(0), J'(0))\in T_{\gamma(0)}M\times T_{\gamma(0)}M}$.

The pair ${(J(0),J'(0))\in T_{\gamma(0)}M\times T_{\gamma(0)}M}$ corresponds to the tangent vector ${\dot{V}(0)}$ at ${s=0}$ to the curve ${V(s)=\frac{\partial\alpha}{\partial t}(s,0)}$ in ${TM}$ (under the identification ${T_{\dot{\gamma}(0)}TM\simeq T_{\gamma(0)}M\times T_{\gamma(0)}M}$ described above [in terms vertical and horizontal subbundles]). Indeed, the curve ${c(s)}$ of footprints of ${V(s)}$ is ${c(s)=\alpha(s,0)}$, so that the tangent vector at ${s=0}$ of ${V(s)}$ is represented by

$\displaystyle (\dot{c}(0), D_c \frac{\partial\alpha}{\partial t}(0,0)) := (J(0), \frac{D}{\partial s}\frac{\partial \alpha}{\partial t}(0,0)) = (J(0), \frac{D}{\partial t}\frac{\partial \alpha}{\partial s}(0,0)) =: (J(0), J'(0))$

Here, the symmetry ${\frac{D}{\partial s}\frac{\partial \alpha}{\partial t}(s,t) = \frac{D}{\partial t} \frac{\partial \alpha}{\partial s}(s,t)}$ of the Levi-Civitta connection was used.

Similarly, the pair ${(J(t), J'(t))}$ represents the tangent vector ${\dot{V}(t)}$ at ${s=0}$ to the curve ${s\mapsto \frac{\partial \alpha}{\partial t}(s,t)=\varphi_t\circ V(s)}$. Therefore, ${(J(t), J'(t))}$ represents

$\displaystyle \dot{V}(t)=D\varphi_t(\dot{V}(0))=D\varphi_t(\dot{\gamma}(0)).$

In summary, Jacobi fields are intimately related to the first derivatives of geodesic flows:

Proposition 1 The image of the tangent vector ${(v_1,v_2)\in T_vTM}$ under the derivative ${D_v\varphi_t}$ of the geodesic flow is the tangent vector ${(J(t), J'(t))\in T_{\varphi_t(v)}TM}$ where ${J}$ is the (unique) Jacobi field with initial data ${(J(0), J'(0))=(v_1, v_2)}$ along the (unique) geodesic ${\gamma}$ with ${\dot{\gamma}(0):=v}$.

2.2. Perpendicular Jacobi fields and invariant subbundles

A concrete example of Jacobi field along a geodesic ${\gamma}$ is ${J(t)=(a+bt)\dot{\gamma}(t)}$: indeed, in this context, ${R(J,\dot{\gamma})\dot{\gamma}=0}$ and ${J''=0}$, so that Jacobi’s equation is trivially verified. Geometrically, this Jacobi field correspond to a trivial variation ${\alpha}$ of the geodesic ${\gamma}$ where the initial point ${\gamma(0)}$ moves along ${\gamma}$ and/or the speed of the parametrization of ${\gamma}$ changes, i.e., ${\alpha(s,t)=\gamma(as+bt)}$.

In general, a Jacobi field ${J}$ along a geodesic ${\gamma}$ that is tangent to ${\gamma}$ has the form ${J(t)=(a+bt)\dot{\gamma}}$ for some ${a,b\in\mathbb{R}}$: in fact, for ${J(t)=j(t)\dot{\gamma}(t)}$ with ${j(t)\in\mathbb{R}}$, one has ${R(J,\dot{\gamma})\dot{\gamma}=0}$, so that Jacobi’s equation reduces to ${J''=0}$, i.e., ${j''(t)=0}$ for all ${t}$.

Hence, a Jacobi field ${J}$ 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 ${J}$ along a geodesic ${\gamma}$ has the following geometrical properties:

• the component ${J'}$ of ${(J,J')}$ makes constant angle with ${\gamma}$, i.e., the quantity ${ \langle J',\dot{\gamma}\rangle}$ is constant;
• if both components of ${(J,J')}$ are orthogonal to ${\gamma}$ at some point, then they stay orthogonal all along ${\gamma}$, i.e., if ${J(t_0)\perp\dot{\gamma}(t_0)}$ and ${J'(t_0)\perp\dot{\gamma}(t_0)}$ for some ${t_0}$, then ${J(t)\perp\dot{\gamma}(t)}$ for all ${t}$;

We say that a Jacobi field ${J}$ along a geodesic ${\gamma}$ is a perpendicular Jacobi field whenever both components of ${(J,J')}$ are orthogonal to ${\gamma}$.

From the properties of Jacobi fields discussed above, we see that any Jacobi field ${J}$ along a geodesic ${\gamma}$ has a decomposition

$\displaystyle J = J_{\parallel}+J_{\perp}$

where ${J_{\perp}}$ is a perpendicular Jacobi field and ${J_{\parallel}}$ is a Jacobi field tangent to ${\gamma}$.

After this little digression on Jacobi fields, let us use them to introduce relevant invariant subbundles under the first derivative of a geodesic flow ${\varphi_t}$.

We begin by recalling that the norm of a tangent vector ${v\in TM}$ stays constant along its ${\varphi_t}$-orbit, i.e., ${\varphi_t}$ preserves the energy hypersurfaces ${\{v\in TM: \|v\|=E\}}$ (for each ${E\geq 0}$). In particular, the first derivative ${D\varphi_t}$ of the geodesic flow ${\varphi_t}$ preserves the tangent bundle ${TT^1M}$ (to the unit tangent bundle ${T^1M}$ of M).

We affirm that, under the identification ${T_v TM\simeq T_p M\times T_p M}$ for ${v\in T_p M}$, the fiber ${T_v T^1M}$ (of the subbundle ${TT^1M}$ of ${TTM}$) corresponds to the set of pairs ${(w_0,w_1)}$ with ${w_1\perp v}$.

In fact, note that an element ${(w_0,w_1)}$ of ${T_vT^1M}$ is tangent at ${s=0}$ to a variation of geodesics ${\alpha(s,t)}$ parametrized by arc-length, i.e.,

$\displaystyle \left\|\frac{\partial\alpha}{\partial t}(s,t)\right\|=1$

for all ${s,t}$, such that the geodesic ${\gamma(t)=\alpha(0,t)}$ satisfies ${\dot{\gamma}(0)=v}$ and the Jacobi field ${J}$ corresponding to ${\alpha(s,t)}$ verifies ${(J(0), J'(0))=(w_0,w_1)}$.

The desired property ${J'(0)=w_1\perp v=\dot{\gamma}(0)}$ now follows from the following calculation:

$\displaystyle \begin{array}{rcl} 0 &=& \frac{\partial D}{\partial s}\left\|\frac{\partial\alpha}{\partial t}\right\|^2(0,0) = 2\langle\frac{D^2\alpha}{\partial s\partial t}, \frac{\partial \alpha}{\partial t}\rangle(0,0) \\ &=& 2 \langle\frac{D^2\alpha}{\partial t\partial s}, \frac{\partial \alpha}{\partial t}\rangle(0,0) = 2\langle J'(0), \dot{\gamma}(0)\rangle \end{array}$

The invariant subbundle ${TT^1M}$ itself admits a decomposition into two invariant subbundles, namely,

$\displaystyle TT^1M =\mathbb{R}\dot{\varphi}\oplus \dot{\varphi}^{\perp}$

where ${\dot{\varphi}}$ is the vector field generating the geodesic flow ${\varphi_t}$ and ${\dot{\varphi}^{\perp}}$ is the orthogonal complement of ${\dot{\varphi}}$. In fact, under the identification ${T_v TM\simeq T_p M\times T_p M}$ for ${v\in T_p M}$, the vector ${\dot{\varphi}(v)}$ is ${(v,0)}$ and the elements of ${\dot{\varphi}^{\perp}}$ have the form ${(w_0,w_1)}$ with ${w_0\perp v}$, ${w_1\perp v}$. In particular, the ${D\varphi_t}$-invariance of ${\dot{\varphi}^{\perp}}$ follows from the fact (mentioned above) that a Jacobi field ${J}$ satisfying ${J(t_0)\perp v}$ and ${J'(t_0)\perp v}$ for some ${t_0}$ is a perpendicular Jacobi field (i.e., ${J(t)\perp v}$ and ${J'(t)\perp v}$ for all ${t}$).

In summary, the action of ${D\varphi_t}$ on ${TT^1M}$ has two complementary invariant subbundles, namely, the span of the vector field ${\dot{\varphi}}$ generating the geodesic flow and its orthogonal ${\dot{\varphi}^{\perp}}$ consisting of perpendicular Jacobi fields. Since ${D\varphi_t}$ acts isometrically in the direction of ${\dot{\varphi}}$, our task is reduced to study the action of ${D\varphi_t}$ on perpendicular Jacobi fields.

2.3. Matrix Jacobi and Ricatti equations

We want to describe the matrix of ${D\varphi_t}$ acting on the vector space of perpendicular Jacobi fields. For this sake, let ${e_1=\dot{\gamma}(0), e_2, \dots, e_n}$ be an orthonormal basis for the tangent space of ${\gamma(0)}$, and denote by ${e_1(t)=\dot{\gamma}(t), e_2(t), \dots, e_n(t)}$ the parallel transport of this orthonormal basis along the geodesic ${\gamma(t)}$.

Define the matrix ${\mathcal{R}(t)}$ whose entries are

$\displaystyle \mathcal{R}_{jk}(t)=\langle R(e_j(t), e_1(t))e_1(t), e_k(t)\rangle$

where ${R}$ is the Riemannian curvature tensor.

Note that any Jacobi field ${J}$ along ${\gamma}$ can be written as ${J(t)=\sum\limits_{k=1}^n y^k(t) e_k(t)}$. In this setting, Jacobi equation becomes

$\displaystyle \frac{d^2 y^k}{dt}+ \sum\limits_{j=1}^n y^j(t) \mathcal{R}_{jk}(t)=0$

and, as usual, a solution is determined by the values ${y^k(0)}$ and ${\frac{dy^k}{dt}(0)}$.

We can write solutions of the Jacobi equation above in a practical way by considering a matrix solution ${\mathcal{J}}$ of the matrix Jacobi equation:

$\displaystyle \mathcal{J}''(t)+\mathcal{R}(t)\mathcal{J}(t)=0$

If ${\mathcal{J}}$ is non-singular, the matrix

$\displaystyle U:=\mathcal{J}'\mathcal{J}^{-1}$

satisfies the matrix Ricatti equation

$\displaystyle U'+U^2+\mathcal{R}=0$

Remark 2 The matrix ${\mathcal{U}}$ is symmetric if and only if one has

$\displaystyle \omega_{\mathbb{R}^{2n}}((J_i, J_i'), (J_j, J_j'))=0$

for any two columns ${J_i}$ and ${J_j}$ of ${\mathcal{J}}$. Here, ${\omega_{\mathbb{R}^{2n}}}$ is the standard symplectic form ${\omega_{\mathbb{R}^{2n}}((x,y),(z,w))=\sum\limits_{i=1}^n (x_i w_i - y_i z_i)}$ of ${\mathbb{R}^{2n}=\mathbb{R}^n\times\mathbb{R}^n}$.

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 ${M}$ be a negatively curved manifold. Let ${0\leq\tau\leq 1}$ and consider ${\gamma:[-\tau,\tau]\rightarrow M}$ a geodesic. Suppose that ${\kappa:[-\tau,\tau]\rightarrow \mathbb{R}_+}$ is a Lipschitz function such that, for each ${-\tau\leq t\leq\tau}$, the sectional curvature of any plane containing ${\dot{\gamma}(t)}$ is greater than or equal to ${-\kappa(t)^2}$, and denote by ${u:[-\tau,\tau]\rightarrow\mathbb{R}_+}$ the solution of Ricatti’s equation

$\displaystyle u'+u^2=\kappa^2$

with initial data ${u(-\tau)=0}$. Then, the first derivative of the geodesic flow ${\varphi_t}$ at time ${\tau}$ satisfies the estimate

$\displaystyle \|D\varphi_{\tau}(\dot{\gamma}(0))\|\leq 1 + 2(1+u(0))^2\left(1+\sqrt{1+u(\tau)^2}\right) \exp\left(\int_0^{\tau} u(s)\,ds\right)$

From our discussion so far, the task of estimating the norm ${\|D\varphi_{\tau}(\dot{\gamma}(0))\|}$ is equivalent to provide bounds for ${\|(J(\tau), J'(\tau))\|_{Sas}}$ in terms of ${\|(J(0), J'(0))\|_{Sas}}$ where ${J(t)}$ is a perpendicular Jacobi field along ${\gamma}$ (cf. Proposition 1 and Subsection 2).

We begin by estimating these quantities for two special subclasses of perpendicular Jacobi fields defined as follows. Let ${\mathcal{X}(t)}$ and ${\mathcal{Y}(t)}$ be the (fundamental) solutions of the matrix Jacobi equation

$\displaystyle \mathcal{J}''(t)+\mathcal{R}(t)\mathcal{J}(t)=0$

with initial data ${\mathcal{X}(0)=\textrm{Id}=\mathcal{Y}(0)}$ and ${\mathcal{X}'(-\tau)=0=\mathcal{Y}(\tau)}$. Note that, by definition, all Jacobi fields ${X(t)}$ with ${X'(-\tau)=0}$, resp. all Jacobi fields ${Y(t)}$ with ${Y(\tau)=0}$, have the form ${X(t)=\mathcal{X}(t)\cdot X(0)}$, resp. ${Y(t)=\mathcal{Y}(t)\cdot Y(0)}$, i.e., they are obtained by applying the matrices ${\mathcal{X}(t)}$, resp. ${\mathcal{Y}(t)}$, to a vector ${X(0)}$, resp. ${Y(0)}$. In this setting, the “other” component ${X'(t)}$, resp. ${Y'(t)}$ (of the Jacobi field ${X(t)}$, resp. ${Y(t)}$, viewed as a tangent vector to ${TM}$) can be recovered by applying the matrix ${\mathcal{U}(t):=\mathcal{X}'(t)\mathcal{X}(t)^{-1}}$, resp. ${\mathcal{V}(t):= \mathcal{Y}'(t) \mathcal{Y}(t)^{-1}}$, to ${X(t)}$, resp. ${Y(t)}$.

Remark 3 Very roughly speaking, the idea behind the choice of the subclasses ${X(t)}$ and ${Y(t)}$ is that ${Y(t)}$ are Jacobi fields belonging to a certain “stable cone” and ${X(t)}$ 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 ${(Y(t), Y'(t))}$ associated to Jacobi fields ${Y(t)}$ as above do not growth in forward time.

Lemma 3 Let ${Y(t)}$ be a perpendicular Jacobi field along ${\gamma}$ such that ${Y(\tau)=0}$. Then,

$\displaystyle \|Y'(0)\|\geq \|Y(0)\|/\tau\geq \|Y'(\tau)\|$

In particular,

$\displaystyle \|(Y(\tau), Y'(\tau)\|_{Sas}=\|Y'(\tau)\|\leq \|Y'(0)\|\leq \|(Y(0), Y'(0))\|_{Sas}$

Proof: One of the consequences of negative sectional curvatures along ${\gamma}$ is the fact that the functions ${\|J(t)\|}$ and ${\|J(t)\|^2}$ are strictly convex for any perpendicular Jacobi field ${J(t)}$ (see, e.g., Eberlein’s survey).

In our context, this implies that ${\|Y\|}$ is a (strictly) convex function decreasing from ${\|Y(0)\|}$ to ${0}$ in the interval ${[0,\tau]}$. Therefore,

$\displaystyle -\|Y\|'(0)\geq \|Y(0)\|/\tau\geq -\|Y\|'(\tau-)$

Since ${\|Y\|'=\langle Y', Y/\|Y\|\rangle}$ and ${Y(t)=Y'(\tau)(t-\tau)+o(\tau-t)}$ for ${t}$ close to ${\tau}$ (because ${Y(\tau)=0}$), we deduce that

$\displaystyle \|Y'(0)\|\geq -\|Y\|'(0)\geq \|Y(0)\|/\tau\geq -\|Y\|'(\tau-)=\|Y'(\tau)\|$

This completes the proof of the lemma. $\Box$

Our second lemma says that the the growth in forward time of tangent vectors ${(X(t), X'(t))}$ associated to Jacobi fields ${X(t)}$ as above is reasonably controlled in terms of the solution ${u}$ of Ricatti’s equation ${u'+u^2=\kappa^2}$ with ${u(-\tau)=0}$ (where ${\kappa}$ is the Lipschitz function controlling some sectional curvatures of ${M}$).

Lemma 4 Let ${X}$ be a perpendicular Jacobi field along ${\gamma}$ with ${X'(-\tau)=0}$. Then,

$\displaystyle \|(X(\tau), X'(\tau))\|_{Sas}\leq \sqrt{1+u(\tau)^2}\exp\left(\int_0^{\tau} u(s) \, ds\right) \|(X(0),X'(0))\|_{Sas}$

Proof: By definition, ${X'(t)=\mathcal{U}(t)\cdot X(t)}$. Thus, ${\|X'(t)\|\leq \|\mathcal{U}(t)\|\cdot\|X(t)\|}$ and, a fortiori,

$\displaystyle \|(X(t), X'(t))\|_{Sas}=\sqrt{\|X(t)\|^2+\|X'(t)\|^2}\leq \sqrt{1+\|\mathcal{U}(t)\|^2}\|X(t)\|$

On the other hand, since ${\|X\|'(t)=\langle X'(t), X(t)/\|X(t)\|\rangle}$, we see that ${\|X\|'(t)\leq \|X'(t)\|\leq \|\mathcal{U}(t)\|\cdot\|X(t)\|}$ and, hence,

$\displaystyle \|X(\tau)\|\leq \exp\left(\int_0^{\tau} \|\mathcal{U}(s)\|\,ds\right)\|X(0)\|$

These inequalities show that the proof of the lemma is complete once we can prove that ${\|\mathcal{U}(t)\|\leq u(t)}$ for all ${t\in[0,\tau]}$.

In this direction, let us observe that the matrix ${\mathcal{U}}$ is symmetric because it verifies (in a trivial way) the condition of Remark 2. Therefore, the norm of ${\mathcal{U}}$ is given by the expression

$\displaystyle \|\mathcal{U}(t)\|=\sup\limits_{\|e\|=1}|\langle\mathcal{U}(t)e,e\rangle|$

where ${e\in\mathbb{R}^{\textrm{dim}(M)-1}}$ ranges from all unit vectors. In particular, our task is reduced to show that

$\displaystyle |u_e(t)|\leq u(t)$

for all unit vectors ${e}$, where ${u_e(t):=\langle\mathcal{U}(t)e,e\rangle}$.

From the matrix Ricatti equation, we see that

$\displaystyle u_e'(t)=\langle\mathcal{U}'(t)e,e\rangle=\langle\mathcal{R}(t)e,e\rangle - \langle\mathcal{U}(t)^2 e,e\rangle$

Since the Lipschitz function ${\kappa}$ controls the sectional curvatures (of planes containing ${\dot{\gamma}}$) along ${\gamma}$ and the matrix ${\mathcal{U}}$ is symmetric, we can estimate the right-hand side of the previous inequality as

$\displaystyle u_e'(t) = \langle\mathcal{R}(t)e,e\rangle - \langle\mathcal{U}(t)^2 e,e\rangle\leq \kappa(t)^2 - \langle \mathcal{U}(t)e, \mathcal{U}(t)e\rangle$

On the other hand, since ${e}$ is a unit vector, the Cauchy-Schwarz inequality implies that ${u_e(t)^2=\langle \mathcal{U}(t)e, e\rangle^2\leq \langle \mathcal{U}(t)e, \mathcal{U}(t)e\rangle}$. Therefore, the right-hand side of the previous inequality is bounded by

$\displaystyle u_e'(t)\leq \kappa(t)^2 - \langle \mathcal{U}(t)e, \mathcal{U}(t)e\rangle\leq \kappa(t)^2 - u_e(t)^2$

From this differential inequality and the facts that ${u_e(-\tau)=0=u(-\tau)}$ and ${u'=\kappa^2-u^2}$, we can easily deduce that ${u_e(t)\leq u(t)}$ for all ${|t|\leq\tau}$ from the standard continuity argument.

Finally, we can complete the proof of the lemma by observing that the symmetric matrix ${\mathcal{U}}$ is positive definite for ${-\tau: this follows from the facts that ${\mathcal{U}(-\tau)=0}$ and ${\mathcal{U}(t)}$ satisfies matrix Ricatti equation associated to a negatively curved manifold ${M}$ (cf. Eberlein’s book). Therefore, ${u_{e}(-\tau)=0}$ and ${u_e(t)>0}$ for all ${-\tau and all unit vector ${e}$, so that

$\displaystyle |u_e(t)|=u_e(t)\leq u(t)$

as desired. $\Box$

Once we know how to control the growth of ${(Y(t),Y'(t))}$ and ${(X(t),X'(t))}$ for Jacobi fields ${Y(t)}$ and ${X(t)}$ as above, the idea to estimate the growth of ${(J(t), J'(t))}$ for an arbitrary perpendicular Jacobi field ${J(t)}$ (thus completing the proof of Theorem 2) is to produce a decomposition of the form

$\displaystyle (J(t), J'(t))=(X(t),X'(t))+(Y(t), Y'(t))$

where ${X'(-\tau)=0=Y(\tau)}$ and the norms of ${(X(0),X'(0))}$ and ${(Y(0), Y'(0))}$ are controlled in terms of the norm of ${(J(0), J'(0))}$.

For this sake, define

$\displaystyle v:=J(0) \quad \textrm{and} \quad w:=(\mathcal{U}(0)-\mathcal{V}(0))^{-1}(J'(0)-\mathcal{U}(0)J(0)))$

and we set ${X(t):=\mathcal{X}(t)(v+w)}$ and ${Y(t):=\mathcal{Y}(t)(-w)}$.

First, note that the vector ${w}$ is well-defined, i.e., the matrix ${\mathcal{U}(0)-\mathcal{V}(0)}$ is invertible. Indeed, we already saw that the matrices ${\mathcal{U}(0)}$ and ${\mathcal{V}(0)}$ are symmetric (because they satisfy (in a trivial way) the condition of Remark 2) and that the matrix ${\mathcal{U}(0)}$ is positive definite (because ${\mathcal{U}(t)}$ satisfies matrix Ricatti equation, the manifold ${M}$ is negatively curved and ${\mathcal{U}(-\tau)=0}$ imply that ${\mathcal{U}(t)}$ is positive-definite for ${-\tau (cf. Eberlein’s book). Furthermore, all eigenvalues of the matrix ${\mathcal{V}(0)}$ are ${\leq -1}$: in fact, any eigenvalue of ${\mathcal{V}(0)}$ has the form ${\langle \mathcal{V}(0)Y(0), Y(0)\rangle=\langle Y'(0),Y(0)\rangle=\|Y\|'(0)}$ for some unit vector ${Y(0)}$, and ${\|Y\|'(0)\leq-1/\tau\leq -1}$ because ${\|Y\|}$ is a convex function (see, e.g., Eberlein’s survey) decreasing from ${\|Y(0)\|=1}$ to ${0}$ in the interval ${[0,\tau]}$ (with ${0<\tau\leq 1}$). Therefore, the matrix ${\mathcal{U}(0)-\mathcal{V}(0)}$ is a symmetric matrix whose eigenvalues are ${\geq 1}$ and, hence, ${\mathcal{U}(0)-\mathcal{V}(0)}$ is an invertible matrix satisfying

$\displaystyle \|(\mathcal{U}(0)-\mathcal{V}(0))^{-1}\|\leq 1$

Secondly, we claim that the Jacobi fields ${X(t)}$ and ${Y(t)}$ give the desired decomposition. In fact, since ${J(t)}$, ${X(t)}$ and ${Y(t)}$ are Jacobi fields, our claim follows from the facts that ${J(0)=(v+w)-w}$ and ${J'(0)=\mathcal{U}(0)(v+w)+\mathcal{V}(0)(-w)}$.

Finally, let us estimate the (Sasaki) norms of ${(X(0), X'(0))}$ and ${(Y(0), Y'(0))}$ in terms of ${(J(0), J'(0))}$. We begin by observing that it suffices to estimate the Sasaki norm of ${(X(0), X'(0)}$ because

$\displaystyle \|(Y(0), Y'(0))\|_{Sas}\leq \|(J(0), J'(0))\|_{Sas}+\|(X(0), X'(0))\|_{Sas}$

On the other hand, the (Sasaki) norm of ${(X(0), X'(0))}$ is not difficult to bound:

$\displaystyle \|(X(0), X'(0)\|_{Sas}\leq (\|v\|+\|w\|)\sqrt{1+\|\mathcal{U}(0)\|^2}$

Since ${\|\mathcal{U}(0)\|\leq u(0)}$ (cf. the proof of Lemma 4) and ${\|(\mathcal{U}(0)-\mathcal{V}(0))^{-1}\|\leq 1}$, we can estimate the right-hand side of the previous inequality by

$\displaystyle \begin{array}{rcl} \|(X(0), X'(0)\|_{Sas}&\leq& (\|v\|+\|w\|)\sqrt{1+\|\mathcal{U}(0)\|^2}\\ &\leq& (\|J(0)\|+\|J'(0)\|+u(0)\|J(0\|))\sqrt{1+u(0)^2} \\ &\leq& \sqrt{2}(1+u(0)^2)(\|J(0)\|+\|J'(0)\|) \\ &=&\sqrt{2}(1+u(0)^2)\|(J(0),J'(0))\|_{Sas} \end{array}$

By putting together these estimates of the Sasaki norms of ${(X(0), X'(0))}$ and ${(Y(0), Y'(0))}$ and Lemmas 3 and 4, we deduce that

$\displaystyle \begin{array}{rcl} \|(J(\tau), J'(\tau))\|_{Sas}&\leq& \|(X(\tau), X'(\tau))\|_{Sas}+\|(Y(\tau), Y'(\tau))\|_{Sas} \\ &\leq & \sqrt{1+u(\tau)^2}\, e^{\int_0^{\tau} u(s)\,ds} \, \|(X(0), X'(0))\|_{Sas} + \|(Y(0), Y'(0))\|_{Sas} \\ &\leq & (1+\sqrt{1+u(\tau)^2}\, e^{\int_0^{\tau} u(s)\,ds}) \, \|(X(0), X'(0))\|_{Sas} + \|(J(0), J'(0))\|_{Sas} \\ &=& (1+\sqrt{2}(1+u(0)^2)(1+\sqrt{1+u(\tau)^2})\, e^{\int_0^{\tau} u(s)\,ds})\|(J(0), J'(0))\|_{Sas} \end{array}$

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 ${N}$ be the quotient ${N=M/\Gamma}$ of a contractible, negatively curved, possibly incomplete, Riemannian manifold ${M}$ by a subgroup ${\Gamma}$ of isometries of ${M}$ acting freely and properly discontinuously. Denote by ${\overline{N}}$ the metric completion of ${N}$ and ${\partial N:=\overline{N}-N}$ the boundary of ${N}$.Suppose that:

• (I) the universal cover ${M}$ of ${N}$ is geodesically convex, i.e., for every ${p,q\in M}$, there exists an unique geodesic segment in ${M}$ connecting ${p}$ and ${q}$.
• (II) the metric completion ${\overline{N}}$ of ${(N,d)}$ is compact.
• (III) the boundary ${\partial N}$ is volumetrically cusplike, i.e., for some constants ${C>1}$ and ${\nu>0}$, the volume of a ${\rho}$-neighborhood of the boundary satisfies

$\displaystyle \textrm{Vol}(\{x\in N: d(x,\partial N)<\rho\})\leq C \rho^{2+\nu}$

for every ${\rho>0}$.

• (IV) ${N}$ has polynomially controlled curvature, i.e., there are constants ${C>1}$ and ${\beta>0}$ such that the curvature tensor ${R}$ of ${N}$ and its first two derivatives satisfy the following polynomial bound

$\displaystyle \max\{\|R(x)\|,\|\nabla R(x)\|,\|\nabla^2 R(x)\|\}\leq C d(x,\partial N)^{-\beta}$

for every ${x\in N}$.

• (V) ${N}$ has polynomially controlled injectivity radius, i.e., there are constants ${C>1}$ and ${\beta>0}$ such that

$\displaystyle \textrm{inj}(x)\geq (1/C) d(x,\partial N)^{\beta}$

for every ${x\in N}$ (where ${inj(x)}$ denotes the injectivity radius at ${x}$).

• (VI) The first derivative of the geodesic flow ${\varphi_t}$ is polynomially controlled, i.e., there are constants ${C>1}$ and ${\beta>0}$ such that, for every infinite geodesic ${\gamma}$ on ${N}$ and every ${t\in [0,1]}$:

$\displaystyle \|D_{\stackrel{.}{\gamma}(0)}\varphi_t\|\leq C d(\gamma([-t,t]),\partial N)^{\beta}$

Then, the Liouville (volume) measure ${m}$ of ${N}$ is finite, the geodesic flow ${\varphi_t}$ on the unit cotangent bundle ${T^1N}$ of ${N}$ is defined at ${m}$-almost every point for all time ${t}$, and the geodesic flow ${\varphi_t}$ is non-uniformly hyperbolic (in the sense of Pesin’s theory) and ergodic.

Actually, the geodesic flow ${\varphi_t}$ is Bernoulli and, furthermore, its metric entropy ${h(\varphi_t)}$ is positive, finite and ${h(\varphi_t)}$ is given by Pesin’s entropy formula (i.e., it is equal to the sum of positive Lyapunov exponents of ${\varphi_t}$ counted with multiplicities).

More precisely, our plan for the rest of this post is to show the non-uniform hyperbolicity of the geodesic flow ${\varphi_t}$ described in the statement above. Then, we will leave the proof of the ergodicity of ${\varphi_t}$ (via Hopf’s argument) for the next post of this series.

We start by noticing that ${N}$ has finite ${m}$-volume: this is an easy consequence of the compactness of ${\overline{N}}$ (assumption (II)) and the volumetrically cusp-like assumption (III) on ${\partial N}$.

Next, let us check that the geodesic flow ${\varphi_t}$ in the statement of Burns-Masur-Wilkinson ergodicity criterion is defined for all time for almost every initial data ${v\in T^1N}$. For this sake, denote by ${\pi:T^1N\rightarrow N}$ the natural projection and set

$\displaystyle U_{\rho}:=\{v\in T^1N: d(\pi(v),\partial N)<\rho\}$

and

$\displaystyle S^+(\rho):=\{v\in T^1N: \varphi_t(v)\in U_{\rho} \textrm{ for some } 0\leq t\leq 1\}$

By definition,

$\displaystyle \{v\in T^1N: \varphi_t(v) \textrm{ is not defined for some } 0\leq t\leq 1\}\subset \bigcap\limits_{\rho>0} S^+(\rho)$

and, a fortiori,

$\displaystyle \{v\in T^1N: \varphi_t(v) \textrm{ is not defined for some } t\in\mathbb{R}\}\subset \bigcup\limits_{k\in\mathbb{Z}}\varphi_{-k}\left(\bigcap\limits_{\rho>0} S^+(\rho)\right)$

In particular, since the geodesic flow ${\varphi_t}$ preserves the measure ${m}$, our task of showing that ${\varphi_t}$ is defined for all time for almost every initial data is reduced to prove that ${\bigcap\limits_{\rho>0} S^+(\rho)}$ has zero ${m}$-measure.

In order to compute the ${m}$-measure of ${\bigcap\limits_{\rho>0} S^+(\rho)}$, let us estimate the ${m}$-measure of ${S^+(\rho)}$ for each ${0<\rho<1}$ along the following lines. Note that

$\displaystyle S^+(\rho)\subset \bigcup\limits_{k=0}^{\lfloor1/\rho\rfloor} V_{k}(\rho)$

where ${V_k(\rho)}$ consists into the unit tangent vectors ${v\in T^1N}$ flowing into ${U_{\rho}}$ for some time between ${k\rho}$ and ${(k+1)\rho}$. By definition, ${\varphi_{(k+1)\rho}(V_k(\rho)) \subset U_{2\rho}}$, so that

$\displaystyle m(V_k(\rho))\leq m(U_{2\rho})\leq C\rho^{2+\nu}$

where ${\nu>0}$. Here, we used the fact that ${m}$ is ${\varphi_t}$-invariant (for the first inequality) and the assumption (III) (for the second inequality). It follows that

$\displaystyle m(S^+(\rho))\leq C\frac{1}{\rho}\rho^{2+\nu} = C\rho^{1+\nu}$

for all ${0<\rho<1}$. Hence, ${\bigcap\limits_{\rho>0} S^+(\rho)}$ has zero ${m}$-measure and ${\varphi_t}$ 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 ${\varphi_t}$ at almost every point: in fact, the weaker condition ${m(U_{\rho})\leq C\rho^{1+\nu}}$ 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 ${\varphi_t}$.

Now, let us show that the geodesic flow ${\varphi_t}$ 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 ${\varphi_t}$ are well-defined (at almost every point): by Oseledets multiplicative ergodic theorem, it suffices to check the ${\log}$-integrability of the derivative cocycles ${D\varphi_1}$ and ${D\varphi_{-1}}$ associated to the time-${1}$ and time-${(-1)}$maps ${\varphi_1}$ and ${\varphi_{-1}}$, that is,

$\displaystyle \int_{T^1N}\log^+\|D\varphi_{\pm1}\| dm<\infty$

By symmetry (or reversibility of the geodesic flow), we have to consider only the ${\log}$-integrability of ${D\varphi_1}$. We estimate the integral above for ${D\varphi_1}$ by noticing that

$\displaystyle \begin{array}{rcl} \int_{T^1N}\log^+\|D\varphi_{1}\| dm &\leq& \int_{T^1N-U_1}\log^+\|D\varphi_{1}\| dm \\ &+& \sum\limits_{n\in\mathbb{N}}\int_{S^+(1/n)-S^+(1/(n+1))}\log^+\|D\varphi_{1}\| dm \end{array}$

Since ${T^1N-U_1}$ 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 ${m(S^+(1/n))\leq C/n^{1+\nu}}$ for some ${\nu>0}$ (as a consequence of assumption (III), and, on the other hand, ${\|D\varphi_1\|\leq \frac{C}{\beta}\log(n+1)}$ on ${S^+(1/n)-S^+(1/(n+1))}$ by assumption (VI), so that

$\displaystyle \sum\limits_{n\in\mathbb{N}}\int_{S^+(1/n)-S^+(1/(n+1))}\log^+\|D\varphi_{1}\| dm\leq \frac{C^2}{\beta}\sum\limits_{n\in\mathbb{N}}\frac{\log(n+1)}{n^{1+\nu}}<\infty$

By Oseledets theorem, once we know the ${\log}$-integrability of the derivative cocycle, we have that, for almost every ${v\in T^1N}$, there are ${k(v)\leq 2n-1}$ real numbers

$\displaystyle \lambda_1(v)>\dots>\lambda_{k(v)}(v)$

called Lyapunov exponents and a ${D\varphi_t}$-invariant splitting

$\displaystyle T_vT^1N=\bigoplus\limits_{i=1}^{k(v)}E_i(v)$

into Lyapunov subspaces ${E_i(v)}$ such that, for every ${\xi\in E_i(v)-\{0\}}$,

$\displaystyle \lim\limits_{t\rightarrow\pm\infty}\frac{1}{t}\log\|D_v\varphi_t(\xi)\|=\lambda_i(v)$

In the context of a geodesic flow ${\varphi_t}$, recall that the derivative cocycle ${D\varphi_t}$ preserves the decomposition ${T_vT^1N=\mathbb{R}\dot{\varphi}\oplus\dot{\varphi}^{\perp}}$, and ${D\varphi_t}$ acts isometrically along ${\mathbb{R}\dot{\varphi}}$ and ${D\varphi_t}$ preserves ${\dot{\varphi}^{\perp}}$. This implies that the Lyapunov exponent of ${\varphi_t}$ along ${\mathbb{R}\dot{\varphi}}$ is zero and the derivative cocycle ${D\varphi_t}$ has ${2n-2}$ Lyapunov exponents counted with multiplicity (i.e., we count ${\textrm{dim}(E_i(v))}$-times the Lyapunov exponent ${\lambda_i(v)}$) along ${\dot{\varphi}^{\perp}}$.

Remark 5 In fact, the derivative cocycle ${D\varphi_t}$ preserves a natural symplectic form on ${\dot{\varphi}^{\perp}}$. In particular, the ${2n-2}$ Lyapunov exponents are organized in a symmetric way around the origin in the sense that ${-\lambda}$ is a Lyapunov exponent whenever ${\lambda}$ is a Lyapunov exponent.

By definition, ${\varphi_t}$ is called non-uniformly hyperbolic whenever all Lyapunov exponents along ${\dot{\varphi}^{\perp}}$ (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 ${\varphi_t}$ by exploiting the negative curvature of ${N}$. More concretely, the negative curvature of ${N}$ implies that:

• for any non-trivial perpendicular Jacobi field ${J(t)}$, the functions ${\|J(t)\|}$ and ${\|J(t)\|^2}$ are strictly convex (thanks to Jacobi’s equation);
• for each geodesic ray ${\gamma:(-\infty,0]\rightarrow N}$ and for each ${w\in\dot{\gamma}(0)=v}$, there exists an unique perpendicular Jacobi field ${J_{w,+}}$ along ${\gamma}$ with ${J_{w,+}(0)=w}$ such that

$\displaystyle \|J_{w,+}(t)\|\leq\|w\|$

for all ${t\leq 0}$.

See, e.g., Eberlein’s book for more explanations. In the literature, ${J_{w,+}}$ is called an unstable Jacobi field and it is usually constructed as the limit ${J_{w,+}=\lim\limits_{\tau\rightarrow-\infty}J_{w,+,\tau}}$ where ${J_{w,+,\tau}}$ is the Jacobi field with ${J_{w,+,\tau}(0)=w}$ and ${J_{w,+,\tau}(\tau)=0}$. Similarly, we can define stable Jacobi fields along geodesic rays ${\gamma:[0,+\infty)\rightarrow N}$ 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 ${\varphi_t}$ and Hopf’s argument depend crucially on the features of stable and unstable manifolds). For now, we just need to know that, if ${N}$ is negatively curved and ${\varphi_t(v)}$ is defined for all time at ${v\in T^1N}$, then

$\displaystyle T_vT^1N=E^s(v)\oplus E^0(v)\oplus E^u(v)$

where ${E^0(v):=\mathbb{R}\dot{\varphi}(v)}$,

$\displaystyle E^s(v)=\{(J(0), J'(0)): J(t) \textrm{ is a stable Jacobi field}\}$

and

$\displaystyle E^u(v)=\{(J(0), J'(0)): J(t) \textrm{ is an unstable Jacobi field}\}.$

In other terms, ${\dot{\varphi}(v)^{\perp}=E^s(v)\oplus E^u(v)}$ where ${E^s(v)}$ and ${E^u(v)}$ are ${(n-1)}$-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 ${\varphi_t}$ is a direct consequence of the following lemma relating stable and unstable Jacobi fields to Lyapunov subspaces:

Lemma 6 There exists a ${\varphi_t}$-invariant subset ${\Lambda_0\subset T^1N}$ of full ${m}$-measure such that

$\displaystyle E^s(v)=\bigoplus\limits_{\lambda_i(v)<0} E_i(v) \quad \textrm{and} \quad E^u(v)=\bigoplus\limits_{\lambda_j(v)>0}E_j(v)$

Proof: Denote by ${\Lambda_0}$ the set of unit vectors ${v\in T^1N}$ such that:

• ${\varphi_t(v)}$ is defined for all time ${t\in\mathbb{R}}$;
• the Lyapunov exponents ${\lambda_i(v)}$ and Lyapunov subspaces ${E_i(v)}$ are defined for ${i=1,\dots,k(v)}$;
• ${v}$ is uniformly recurrent under ${\varphi_t}$ in the sense that, for any neighborhood ${U}$ of ${v}$, there exists ${\delta>0}$ such that the sets ${R_{\pm}(T)=\{\pm t\in[0,T]: \varphi_t(v)\in U\}}$ have Lebesgue measure ${\geq\delta T}$ for all ${T}$ sufficiently large.

Note that ${\Lambda_0}$ is ${\varphi_t}$-invariant and it has full ${m}$-measure: our previous discussion showed that the first two conditions hold for almost every ${v\in T^1N}$ and the third condition holds in a full measure subset thanks to Birkhoff’s ergodic theorem.

We affirm that ${\Lambda_0}$ satisfies the conclusions of the lemma. In fact, by the reversibility of the geodesic flow ${\varphi_t}$, it suffices to show that

$\displaystyle E^u(v)=\bigoplus\limits_{\lambda_j(v)>0}E_j(v)$

for all ${v\in\Lambda_0}$.

For this sake, given ${v\in\Lambda_0}$, we fix a neighborhood ${U}$ of ${v}$ and a real number ${\eta>0}$ such that if ${J(t)}$ is an unstable Jacobi field along a geodesic ${\gamma}$ with ${\dot{\gamma}(0)\in U}$, then

$\displaystyle \|J(1)\|\geq (1+\eta)\|J(0)\|$

The choice of ${U}$ and ${\eta}$ is possible because ${N}$ is negatively curved and ${\|J(t)\|}$ is an increasing strictly convex function whose second derivative is controlled by Jacobi’s equation.

Since ${v\in\Lambda_0}$ is uniformly recurrent for ${\varphi_t}$, we have that

$\displaystyle \|J(t+1)\|\geq (1+\eta)\|J(t)\|$

for all ${t\in R_+(T):=\{s\in[0,T]:\varphi_s(v)\in U\}}$. Because ${ v\in\Lambda_0}$, we know that ${R_+(T)}$ has Lebesgue measure ${\geq\delta T}$for some ${\delta>0}$ and for all ${T}$ sufficiently large. Therefore, for any unstable Jacobi field ${J(t)}$ along ${\varphi_t(v)}$, one has

$\displaystyle \|J(T)\|\geq (1+\eta)^{\delta T-1}\|J(0)\|$

for all ${T}$ sufficiently large. It follows from the definitions that

$\displaystyle \lim\limits_{T\rightarrow+\infty}\frac{1}{T}\log\|D_v\varphi_T(\xi)\|\geq\delta\log(1+\eta)>0$

for any ${\xi\in E^u(v)}$, and, hence,

$\displaystyle E^u(v)\subset\bigoplus\limits_{\lambda_j(v)>0} E_j(v)$

Similarly, ${E^s(v)\subset\bigoplus\limits_{\lambda_i(v)<0} E_i(v)}$. Because ${E^s(v)\oplus E^u(v)=\dot{\varphi}(v)^{\perp}}$, these inclusions must be equalities and the proof of the lemma is complete. $\Box$

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 ${\varphi_t}$ is non-uniformly hyperbolic: more concretely, there exists a subset ${\Lambda_0\subset T^1N}$ of full ${m}$-measure such that the ${D\varphi_t}$-invariant splitting

$\displaystyle T_vT^1N=E^s(v)\oplus E^0(v)\oplus E^u(v)$

into the flow direction ${E^0(v)=\mathbb{R}\dot{\varphi}(v)}$ and the spaces ${E^s(v)}$ and ${E^u(v)}$ of stable and unstable Jacobi fields along ${\gamma(t)=\varphi_t(v)}$ have the property that

$\displaystyle 0<\lim\limits_{t\rightarrow\infty}\frac{1}{t}\log\|D_v\varphi_t(\xi^u)\|<\infty \quad \textrm{and} \quad -\infty<\lim\limits_{t\rightarrow\infty}\frac{1}{t}\log\|D_v\varphi_t(\xi^s)\|<0$

for all ${\xi^u\in E^u(v)-\{0\}}$ and ${\xi^s\in E^s(v)-\{0\}}$.