Posted by: matheuscmss | June 24, 2014

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

Last time, we showed the first part of Burns-Masur-Wilkinson ergodicity criterion:

Theorem 1 (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, we proved in the previous post of this series that a geodesic flow ${\varphi_t}$ satisfying the assumptions (II), (III) and (VI) above is non-uniformly hyperbolic with respect to the volume probability measure, and, furthermore, we identified the Oseledets stable and unstable subspaces (cf. the last theorem of this post):

Theorem 2 Under the assumptions (II), (III) and (VI) in Theorem 1 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 ${\xi^s\in E^s(v)-\{0\}}$.

Today, we want to exploit the non-uniform hyperbolicity of ${\varphi_t}$ (and the assumptions (I) to (VI) above) in order to deduce the ergodicity of ${\varphi_t}$ via Hopf’s argument.

For this sake, we organize this post as follows. In the first section, we discuss the geometry of stable and unstable manifolds of ${\varphi_t}$: in particular, we will see that these invariant manifolds form global laminations with useful absolute continuity properties. After that, we describe Hopf’s argument in the second section: from the nice properties of the invariant laminations, we deduce that Birkhoff averages are constant almost everywhere, and, hence, ${\varphi_t}$ is ergodic. Finally, we conclude this post with a remark (inspired by conversations with Y. Coudène and B. Hasselblatt last November 2013) about the deduction of the mixing property for ${\varphi_t}$ from Hopf’s argument.

1. Stable manifolds of certain geodesic flows

1.1. Local (Pesin) stable manifolds for certain geodesic flows

We begin by noticing that a geodesic flow ${\varphi_t}$ satisfying the assumptions (I) to (VI) of Theorem 1 has “nice” local (Pesin) stable and unstable manifolds through almost every point.

The reader with some experience with non-uniformly hyperbolic systems might think that this is an immediate consequence of the so-called Pesin’s theory. However, this is not the case in our setting because the phase space ${T^1N}$ of ${\varphi_t}$ is not assumed to be compact. In other words, we are facing a dilemma: while the non-compactness of ${N}$ is an important point for the applications of Theorem 1 (to moduli spaces equipped with WP metrics), it forbids a naive utilization of Pesin’s theory because of the competition between the dynamical behaviors of ${\varphi_t}$ in compact regions of ${N}$ and near “infinity” ${\partial N}$.

Fortunately, Katok and Strelcyn (with the aid of Ledrappier and Przytycki) developed a generalization of Pesin’s theory where any “well-behaved” dynamics on non-compact phase space is allowed. Furthermore, Katok-Strelcyn successfully applied their version of Pesin’s theory to the study of dynamical billiards.

Very roughly speaking, Katok-Strelcyn say that if the dynamics of the non-uniformly hyperbolic system ${\varphi_t}$ “blows up at most polinomially” at infinity ${\partial N}$, then the hyperbolic (exponential) behavior of ${\varphi_t}$ is strong enough so that Pesin’s theory can be applied (because ${N}$ is “essentially compact” for practical purposes).

Evidently, this is much easier said than done, and, unfortunately, the discussion of the details of Katok-Strelcyn’s generalization of Pesin’s theory is out of the scope of this post. In particular, we will content ourselves in just mentioning the conditions (I) to (VI) in Theorem 1 were set up by Burns-Masur-Wilkinson in such a way that a geodesic flow ${\varphi_t}$ satisfying (I) to (VI) also verifies all the requirements to apply Katok-Strelcyn’s work. Here, even though this is philosophically natural, it is worth to point out that the deduction of the conditions to use Katok-Strelcyn’s technology from (I) to (VI) is far from trivial: indeed, Burns-Masur-Wilkinson do this after studying (in Appendices A and B of their paper) several ${C^3}$ properties of Sasaki metric and ${C^2}$ properties of ${\varphi_t}$.

In summary, Burns-Masur-Wilkinson use (I) to (VI) to ensure that Katok-Strelcyn’s generalization of Pesin’s theory applies in the setting of Theorem 1. As a by-product, they deduce the following statement about the existence and absolute continuity of local (Pesin) stable manifolds (cf. Proposition 3.10 of Burns-Masur-Wilkinson paper).

Theorem 3 (“Pesin stable manifold theorem”) Let ${\varphi_t}$ be the geodesic flow on the unit tangent bundle ${T^1N}$ of a ${n}$-dimensional Riemannian manifold ${N}$ satisfying the conditions (I) to (VI) of Theorem 1. Denote by ${\Lambda_0\subset T^1N}$ the subset of full volume provided by Theorem 2 where ${\varphi_t}$ is non-uniformly hyperbolic.Then, there exists a subset ${\Lambda_1\subset\Lambda_0}$ of full volume, a \textrm{measurable} function ${r:\Lambda_1\rightarrow\mathbb{R}_+}$, and a measurable family

$\displaystyle W^s_{loc}=\{W^s_{loc}(v) : v\in\Lambda_1\}$

of smooth (${C^{\infty}}$) embedded disks ${W^s_{loc}(v)}$ with the following properties. For all ${v\in\Lambda_1}$:

• ${T_v W^s_{loc}(v)=E^s(v)}$, i.e., ${W^s_{loc}(v)}$ is tangent to ${E^s(v)}$;
• ${\varphi_t(W^s_{loc}(v))\subset W^s_{loc}(v)}$ for all ${t\geq 0}$, i.e., ${W^s_{loc}(v)}$ is topologically contracted in forward time by ${\varphi_t}$;
• ${w\in W^s_{loc}(v)}$ if and only if ${d_{Sas}(v,w) and ${\lim\limits_{t\rightarrow+\infty} d_{Sas}(\varphi_t(v),\varphi_t(w))=0}$, i.e., ${W^s_{loc}(v)}$ is local stable manifold (in the sense that it is dynamically characterized as the set of ${w}$ close to ${v}$ whose forward ${\varphi_t}$-orbit approaches the forward ${\varphi_t}$-orbit of ${w}$).

Moreover, the family ${W^s_{loc}}$ is absolutely continuous in the sense that the following “Fubini-like statements” hold.

• given ${Z\subset T^1N}$ a subset of zero volume, one has that the set ${Z\cap W^s_{loc}(v)}$ has zero measure in ${W^s_{loc}(v)}$ (with respect to the induced ${(n-1)}$-dimensional Lebesgue measure on ${W^s_{loc}(v)}$) for almost every ${v\in\Lambda_1}$;
• given a ${C^1}$-embedded ${n}$-dimensional open disk ${D\subset T^1N}$ and ${B\subset D}$ a subset of zero measure (for the induced Lebesgue measure of ${D}$), the set

$\displaystyle Sat_{loc}^s(B):=\bigcup\limits_{\substack{v\in\Lambda_1, \\ W^s_{loc}(v)\cap B\neq\emptyset}} W^s_{loc}(v)$

(obtained by saturating ${B}$ by the local stable manifolds ${W^s_{loc}(v)}$ passing through it) has zero volume in ${T^1N}$.

Finally, the analogous assertions about unstable manifolds are also true.

1.2. Global stable manifolds of certain geodesic flows

The Pesin stable and unstable laminations provided by Theorem 3 are not sufficient to run Hopf’s argument: as it was explained in the first post of this series, the local stable manifolds ${W^s_{loc}(v)}$ could be a priori very short (because their radii ${r(v)}$ vary only measurably with ${v\in\Lambda_1}$ and so one does not expect for uniform lower bounds on ${r(v)}$).

Hence, it is important (for our purposes of using Hopf’s argument) to compare Pesin’s local stable manifolds ${W^s_{loc}}$ with global objects. Here, the key point is to observe that Theorem 2 says that the tangent space of ${W^s_{loc}(v)}$ at ${v}$ is exactly the vector space of stable Jacobi fields along the geodesic ${\varphi_t(v)}$ and, as we will recall in a moment, stable Jacobi fields are naturally related to global objects called stable horospheres.

1.2.1. Stable Jacobi fields and stable horospheres

Let ${N}$ be a Riemannian manifold. Given an unit tangent ${v\in T^1N}$ generating a geodesic ray ${\gamma:[0,\infty)\rightarrow N}$ such that the sectional curvatures of ${N}$ are negative along ${\gamma}$ and ${w\in\dot{\varphi}(v)^{\perp}}$, let us denote by ${J_{-,w}}$ the stable Jacobi field associated to ${w}$: by definition, this is the Jacobi field

$\displaystyle J_{-,w}(t):=\lim\limits_{\tau\rightarrow+\infty}J_{-,w,\tau}(t)$

where ${J_{-,w,\tau}(t)}$ is the Jacobi field satisfying ${J_{-,w,\tau}(0)=w}$ and ${J_{-,w,\tau}(\tau)=0}$.

In terms of the description of Jacobi fields via variations of geodesics, the stable Jacobi fields along ${\gamma}$ are obtained by varying ${\gamma}$ through geodesics ${\beta:[0,+\infty)\rightarrow N}$ such that ${d(\beta(t),\gamma(t))\leq d(\beta(0),\gamma(0))}$ for all ${t\geq 0}$ (that is, ${\beta}$ stays always close to ${\gamma}$ in forward time). These geodesics ${\beta}$ are orthogonal to a family of immersed hypersurfaces of ${N}$ whose lifts to the universal cover ${M}$ of ${N}$ are the so-called stable horospheres.

The stable horospheres can be constructed “by hands” with the aid of Busemann functions as follows.

Let ${N}$ be the quotient ${N=M/\Gamma}$ of a contractible, negatively curved, Riemannian manifold ${M}$ by a subgroup ${\Gamma}$ of isometries of ${M}$ acting freely and properly discontinuously and suppose that the universal cover ${M}$ of ${N}$ is geodesically convex (i.e., ${M}$ satisfies item (I) of Theorem 1).

In this situation, it is possible to show (see, e.g., Proposition 3.5 in Burns-Masur-Wilkinson paper) that given an unit vector ${v\in T^1M}$ generating an infinite geodesic ray ${\gamma_v:[0,+\infty)\rightarrow M}$, the functions ${b^s_{v,t}:M\rightarrow\mathbb{R}}$ given by

$\displaystyle b^s_{v,t}(y)=d(y,\gamma_v(t))-t$

converge (uniformly on compact sets) as ${t\rightarrow+\infty}$ to a ${C^1}$ convex function

$\displaystyle b^s_v:M\rightarrow\mathbb{R}$

called stable Busemann function such that ${\|\textrm{grad}(b^s_v)\|=1}$ and, for every ${y\in M}$, the unit vector ${w^s_v(y):=-\textrm{grad}(b^s_v)(y)}$ defines an infinite geodesic ray ${\gamma_{w^s_v(y)}:[0,+\infty)\rightarrow M}$ with

$\displaystyle d(\gamma_{w^s_v(y)}(t),\gamma_v(t))\leq d(\gamma_v(0),y)$

for all ${t\geq 0}$. In particular, the geodesics ${\gamma_{w^s_v(y)}(t)}$ give variations of ${\gamma}$ leading to stable Jacobi fields.

For each ${t\in\mathbb{R}}$, the level set ${\mathcal{H}_v^s(t)=(b^s_v)^{-1}(t)\subset M}$ is a connected, complete, codimension ${1}$ submanifold of ${M}$ called stable horosphere of level ${t}$. By definition, the geodesics ${\gamma_{w^s_v(y)}}$ are orthogonal to the ${1}$-parameter family ${\mathcal{H}^s_v(t)}$ of stable horospheres (because stable horospheres are leve sets of ${b^s_v}$ and the geodesics ${\gamma_{w^s_v(y)}}$ point in the direction ${w^s_v(y):=-\textrm{grad}(b^s_v)(y)}$ of the gradient).

The submanifold

$\displaystyle W^s(v):=\{w^s_v(y): y\in\mathcal{H}^s_v(0)\}$

of ${T^1M}$ consisting of unit vectors that are orthogonal to the stable horosphere ${\mathcal{H}^s_v(0)}$ of level ${0}$ is called the (global) stable manifold of ${v\in T^1M}$. This nomenclature is justified by the following property (corresponding to Proposition 3.6 in Burns-Masur-Wilkinson paper). In the context of Theorem 1, suppose that the infinite geodesic ray ${\gamma_v:[0,+\infty)\rightarrow M}$ projecting to a forward recurrent geodesic on ${N=M/\Gamma}$ (i.e., after projection to ${N}$, the unit vector ${\dot{\gamma}(0)}$ becomes an accumulation point of the set ${\{\dot{\gamma}(t):t\geq 1\}}$). Then, for any ${y\in M}$, the unit vector ${w=w^s_v(y)\in T^1_y M}$ is tangent to an infinite geodesic ray ${\gamma_w:[0,+\infty)\rightarrow M}$ such that

$\displaystyle \lim\limits_{t\rightarrow+\infty}d(\gamma_v(t), \gamma_w(t+b^s_v(y)))=0$

Furthermore, ${d_{Sas}(\varphi_t(v),\varphi_{t+b^s_v(y)}(w))\rightarrow 0}$ as ${t\rightarrow+\infty}$. In particular, ${\varphi_t(W^s(v))= W^s(\varphi_t(v))}$ (stable manifolds are ${\varphi_t}$-invariant) and ${\lim\limits_{t\rightarrow+\infty} d_{Sas}(\varphi_t(v), \varphi_t(w))=0}$ for all ${w\in W^s(v)}$ (stable manifolds are dynamically characterized by future orbits getting close together).

Remark 1 As usual, by reversing the time (via the symmetry ${\gamma_v(t)=\gamma_{-v}(-t)}$), one can define unstable Jacobi fields, unstable Busemann functions and unstable horospheres.

The following picture (that we already encountered in the last post while discussing Jacobi fields) illustrates the stable and unstable horospheres associated to the vertical geodesic in the hyperbolic plane passing through ${i}$.

Fig 1. Stable and unstable horospheres in the hyperbolic plane.

1.2.2. Geometry of the stable and unstable horospheres

In this subsection, we make a couple of comments on the geometry of stable and unstable horospheres. More precisely, besides explaining the computation of their second fundamental forms from matrix Riccati equations, we will see that the stable and unstable horospheres are mutually transverse in a quantitative way. Of course, this transversality property of horospheres is another important point in Hopf’s argument (as it allows to control the angle between stable and unstable manifolds).

Let ${\gamma:(-\infty,0]\rightarrow M}$ be a geodesic ray such that the sectional curvatures of ${M}$ along ${\gamma}$ are negative. For each ${w\in\dot{\gamma}(0)^{\perp}}$, let us denote by ${J_{+,w}(t)}$ the unstable Jacobi field along ${\gamma}$ with ${J_{+,w}(0)=w}$ (as usual).

Consider the ${1}$-parameter family of matrices (linear operators) ${U_+(t):\dot{\gamma}(t)^{\perp}\rightarrow\dot{\gamma}(t)^{\perp}}$ defined by the formula

$\displaystyle U_+(t)(J_{+,w}(t))=J_{+,w}'(t)$

As we mentioned in this post here, ${U_+(t)}$ are symmetric, positive-definite operators satisfying the matrix Ricatti equation

$\displaystyle U_+' + U_+ +\mathcal{R}=0,$

(i.e., ${\langle a, U_+(t)(a)\rangle = -\langle a, R(a,\dot{\gamma}(t))\dot{\gamma}(t)\rangle -\langle a , U_+(t)^2(a)\rangle}$ for all ${a\in\dot{\gamma}(t)^{\perp}}$).

It is possible to show (cf. Eberlein’s survey) that the operator ${U_+(t)}$ is precisely the second fundamental form at ${\dot{\gamma}(t)}$ of the unstable horosphere ${\mathcal{H}^u_v(t)}$ of level ${t}$.

By reversing the time, we have an analogous operator ${U_-(t)}$ related to stable horospheres.

Note that, by definition, the stable and unstable subspaces ${E^s(v)}$ and ${E^u(v)}$ at an unit vector ${v=\dot{\gamma}(0)}$ defining an infinite geodesic ray ${\gamma:\mathbb{R}\rightarrow M}$ are

$\displaystyle E^u(v)=\{(a,U_+(0)a) : a\in v^{\perp}\} \textrm{ and } E^s(v)=\{(b,U_-(b)) : b\in v^{\perp}\}$

In other terms, we have a ${D\varphi_t}$-invariant splitting

$\displaystyle T_{\mathcal{D}}T^1M = E^s\oplus E^0\oplus E^u$

over the set

$\displaystyle \mathcal{D}:=\{v\in T^1M: v \textrm{ defines an infinite geodesic ray }\gamma:\mathbb{R}\rightarrow M\}$

(where ${E^0=\mathbb{R}\dot{\varphi}}$).

Let us now show that this splitting is locally uniform over ${\mathcal{D}}$.

Proposition 4 There exists a continuous function ${\delta:T^1M\rightarrow\mathbb{R}_+}$ such that the continuous family of conefields

$\displaystyle \mathcal{C}^s(v)=\{(w,w')\in \dot{\varphi}(v)^{\perp}: \langle w, w'\rangle\leq -\delta(v)\|(w,w')\|_{Sas}\}$

and

$\displaystyle \mathcal{C}^u(v)=\{(w,w')\in \dot{\varphi}(v)^{\perp}: \langle w, w'\rangle\geq \delta(v)\|(w,w')\|_{Sas}\}$

meeting only at the origin have the property that

$\displaystyle E^s(v)\subset \mathcal{C}^s(v) \textrm{ and } E^u(v)\subset \mathcal{C}^u(v)$

for all ${v\in\mathcal{D}}$.

Proof: Our task consists in showing that the functions

$\displaystyle \delta^u(v):=\inf\limits_{(w,w')\in E^u(v)-\{0\}} \frac{\langle w,w'\rangle}{\|(w,w')\|_{Sas}^2} \textrm{ and } \delta^s(v):=\inf\limits_{(w,w')\in E^s(v)-\{0\}} -\frac{\langle w,w'\rangle}{\|(w,w')\|_{Sas}^2}$

of ${v\in\mathcal{D}}$ are locally uniformly bounded away from zero.

By symmetry, it suffices to prove that ${\delta^s}$ is locally uniformly bounded from below. For the sake of reaching a contradiction, suppose this is not the case. This means that there are sequences ${v_n\in\mathcal{D}}$, ${\xi_n\in E^s(v_n)-\{0\}}$ with ${\|\xi_n\|_{Sas}=1}$ such that ${v_n\rightarrow v\in\mathcal{D}}$, ${\xi_n\rightarrow\xi=(w,w')}$ and ${\langle w, w'\rangle=0}$.

For each ${n\in\mathbb{N}}$, let ${J_n}$ be the stable Jacobi fields along ${\gamma_{v_n}}$ induced by ${\xi_n}$, and denote by ${J}$ the (limit) Jacobi field along ${\gamma_v}$ induced by ${\xi}$.

On one hand, for each ${n}$, the square ${\|J_n(t)\|^2}$ of the norm of the stable Jacobi field ${J_n(t)}$ is a decreasing function of ${t}$. In particular, since ${\xi_n\rightarrow \xi}$, we deduce that ${\|J(t)\|^2}$ is a non-increasing function of ${t}$.

On the other hand, ${\|J(t)\|^2}$ is a strictly convex function of ${t}$ (because ${J}$ is a perpendicular Jacobi field, cf. Eberlein’s survey).

By putting these two facts together, we see that the function ${t\mapsto \|J(t)\|^2}$ has no critical points. However, ${(\|J\|^2)'(0)=2\langle w,w'\rangle=0}$. This contradiction proves the desired proposition. $\Box$

1.2.3. Absolute continuity of global stable manifolds

Once we have related Pesin’s stable and unstable manifolds ${W^s_{loc}}$ (local objects) to stable and unstable horospheres (global objects), it is not entirely surprising that the absolute continuity properties of Pesin stable manifolds (described in Theorem 3 above) can be “transferred” to horospherical laminations:

Proposition 5 Let ${\varphi_t}$ be the geodesic flow on the unit tangent bundle ${\pi:T^1N\rightarrow N}$ of a ${n}$-dimensional Riemannian manifold ${N}$ satisfying the conditions (I) to (VI) of Theorem 1. Denote by ${\Omega_1\subset T^1M}$ the subset of the unit tangent bundle of the universal cover ${p:M\rightarrow N}$ of ${N}$ consisting of unit vectors ${v\in T^1M}$ projecting into a forward and backward recurrent geodesic ${\gamma_v}$ in ${T^1N}$.Then, there exists a subset ${\Omega_2\subset\Omega_1}$ of full volume such that the stable Busemann functions ${b_v^s:M\rightarrow \mathbb{R}}$ are ${C^{\infty}}$ for all ${v\in\Omega_2}$. Moreover, the leaves of the stable lamination ${W^s=\{W^s(v):v\in\Omega_2\}}$ are ${C^{\infty}}$-submanifolds of ${T^1M}$ diffeomorphic to ${\mathbb{R}^{n-1}}$. Furthermore, the stable horospherical lamination

$\displaystyle \{W^s(v,\delta): v\in\Lambda_2, \delta<\textrm{inj}(\pi(v))\}$

obtained by taking the family of manifolds ${W^s(v,\delta):=\textrm{ connected component of }W^s(v)\cap B_{T^1N}(v,\delta) \textrm{ containing } v}$ through the vectors ${v\in \Lambda_2}$ in the projection ${\Lambda_2=Dp(\Omega_2)}$ of ${\Omega_2}$ to ${T^1N}$ (via ${Dp:T^1M\rightarrow T^1N}$) has the following absolute continuity properties:

• if ${Z\subset T^1N}$ has zero ${m}$-volume, then for ${m}$-almost every ${v\in\Lambda_2}$ and any ${\delta<\textrm{inj}(\pi(v))}$, the set ${Z\cap W^s(v,\delta)}$ has zero ${(n-1)}$-dimensional volume in ${W^s(v,\delta)}$;
• if ${D\subset T^1N}$ is a smooth, embedded, ${n}$-dimensional open disk and ${B\subset D}$ has zero ${n}$-dimensional volume in ${D}$, then for any ${\delta<\frac{1}{2} \inf\limits_{v\in D} \textrm{inj}(\pi(v))}$ one has ${m(Sat^s(B,\delta))=0}$ where

$\displaystyle Sat^s(B,\delta):=\bigcup\limits_{v\in\Lambda_2: W^s(v,\delta)\cap B\neq\emptyset} W^s(v,\delta)$

is the set obtained by saturating ${B}$ with the leaves of the lamination ${W^s(v,\delta)}$.

Finally, a similar statement holds for the corresponding unstable lamination.

Logically, the statement of this proposition is very close to Theorem 3 about the absolute continuity of Pesin stable manifolds, but the crucial point is that we have now an absolutely continuous stable lamination ${W^s}$ whose leaves have radii essentially equal to ${\textrm{inj}(\pi(v))/2}$. In other words, the leaves of the stable lamination ${W^s}$ have a size controlled by the injectivity radius of ${N}$, a global smooth function, instead of the a priori merely measurable function ${r(v)}$ giving the radii of leaves of Pesin’s stable lamination ${W^S_{loc}}$.

The proof of Proposition 5 is not very difficult: it uses the absolute continuity properties of Pesin’s lamination ${W^s_{loc}}$ in Theorem 3 and the “contraction of stable horospheres” (i.e., the fact that the forward dynamics of ${\varphi_t}$ eventually contracts ${W^s(v,\delta)}$ inside ${W^s(\varphi_t(v),r(v))}$), and it occupies two pages in Burns-Masur-Wilkinson paper (cf. the proof of their Proposition 3.11). However, we will skip this point in favor of discussing Hopf’s argument right now.

2. Proof of Theorem 1 via Hopf’s argument

Let ${\varphi_t}$ be a geodesic flow satisfying the assumptions (I) to (VI) of Theorem 1. We want to show that ${\varphi_t}$ is ergodic with respect to the volume measure ${m}$ (with normalized total mass).

By Birkhoff’s ergodic theorem, given a continuous function ${f:T^1N\rightarrow\mathbb{R}}$ with compact support, the Birkhoff ergodic averages

$\displaystyle \frac{1}{T}\int_0^T f(\varphi_t(v)) dt$

converge as ${T\rightarrow\pm\infty}$ to the same limit ${B(f)(v)}$ for ${m}$-almost every ${v\in T^1N}$.

By definition of ergodicity, our task consists in showing that the function ${B(f)(v)}$ is constant ${m}$-almost everywhere.

For this sake, let us define the measurable functions

$\displaystyle f^s(v)=\limsup\limits_{T\rightarrow+\infty}\frac{1}{T}\int_0^T\varphi_t(v)dt$

and

$\displaystyle f^u(v)=\limsup\limits_{T\rightarrow-\infty}\frac{1}{T}\int_0^{T}\varphi_s(v) ds$

Note that, by Birkhoff’s ergodic theorem, there exists a subset ${G\subset T^1N}$ of full ${m}$-measure such that

$\displaystyle f^s(v)=f^u(v)=B(f)(v)$

Moreover, from their definitions, note that the functions ${f^s}$, ${f^u}$ and ${B(f)}$ are ${\varphi_t}$-invariant.

The initial observation in Hopf’s argument is the fact that the function ${f^s}$, resp. ${f^u}$, is constant along the stable manifolds ${W^s(v)}$, resp. unstable manifolds ${W^u(v)}$. In fact, this follows easily from the uniform continuity of the (compactly supported, continuous) function ${f}$ and the fact that ${d(\varphi_t(v), \varphi_t(w))\rightarrow 0}$ as ${t\rightarrow+\infty}$ (resp. ${t\rightarrow-\infty}$) whenever ${w\in W^s(v)}$ (resp. ${w\in W^u(v)}$).

The basic strategy of Hopf’s argument can be summarized as follows. We want to combine this initial observation with the absolute continuity properties of the stable and unstable horospherical laminations to deduce that ${\varphi_t}$ is “locally ergodic” in the sense that every ${v\in T^1N}$ possesses a neighborhood ${U_v}$ such that the restriction ${B(f)}$ to ${U_v}$ is ${m}$-almost everywhere constant.

Of course, since ${T^1N}$ is connected, this local ergodicity property implies that the function ${B(f)}$ is constant ${m}$-almost everywhere, and, a fortiori, ${\varphi_t}$ is ergodic with respect to ${m}$. In other terms, our task is reduced to prove the local ergodicity property stated in the previous paragraph.

In this direction, we fix once and for all ${v\in T^1N}$, we set

$\displaystyle \delta=\delta(v):=\frac{1}{4}\min\{\textrm{inj}(\pi(v)), d(v,\partial N)\},$

and we denote by ${V}$ the ${\delta}$-neighborhood of ${v\in T^1N}$.

Let ${\Lambda_2\subset T^1N}$ be the full ${m}$-volume subset constructed in Proposition 5. For each ${w\in \Lambda_2\cap V}$, we consider the stable leaf ${W^s(w,\delta)}$, we take its iterates under ${\varphi_t}$ for ${|t|<\delta}$, and we saturate the resulting subset ${\varphi_{(-\delta,\delta)}(W^s(w,\delta))=\bigcup\limits_{|t|<\delta} \varphi_t(W^s(w,\delta))}$ with the leaves of the unstable horospherical lamination ${W^u=\{W^u(.,\delta)\}}$ to obtain the subset

$\displaystyle N_{\delta}(w):= Sat^u(\varphi_{(-\delta,\delta)}(W^s(w,\delta)), \delta)$

The construction of ${N_{\delta}(w)}$ is illustrated in the figure below: the subset ${\varphi_{(-\delta,\delta)}(W^s(w,\delta))}$ is marked in blue and some leaves of ${W^u}$ passing through points of ${\varphi_{(-\delta,\delta)}(W^s(w,\delta))}$ are marked in red.

The local ergodicity property stated above is an immediate consequence of the following two claims:

• (a) the restriction of the function ${B(f)}$ to ${N_{\delta}(w)}$ is almost everywhere constant for almost every choice of ${w\in\Lambda_2\cap V}$;
• (b) ${N_{\delta}(w)}$ is essentially open for almost every ${w}$ near ${v}$ in the sense that there exists a neighborhood ${U_v}$ of ${v}$ such that ${N_{\delta}(w)\cap U_v}$ has full volume in ${U_v}$ for almost every choice of ${w\in U_v}$.

We establish the first claim (a) by exploiting the initial observation that Birkhoff averages are constant along stable and unstable manifolds and the absolute continuity properties of the stable and unstable horospherical laminations.

More precisely, let us consider again the full volume subset ${G}$ of ${T^1N}$ where ${f^s(v)=f^u(v)=B(f)(v)}$ (provided by Birkhoff’s ergodic theorem).

By absolute continuity property of ${W^s}$ (cf. the first item of conclusion of Proposition 5), for almost every ${w\in\Lambda_2\cap V}$, the intersection ${G\cap W^s(w,\delta)}$ has full volume in ${W^s(w,\delta)}$. We affirm that ${B(f)|_{N_{\delta}(w)}}$ is almost everywhere constant for any such ${w}$.

In fact, ${f^s}$ takes a constant value ${a:=f^s(w)}$ on ${W^s(w,\delta)}$. Moreover, since ${f^s=f^u}$ on ${G}$, we also have that ${f^u}$ takes the constant value ${a}$ on ${G\cap W^s(w,\delta)}$. By combining this fact with the ${\varphi_t}$-invariance of ${f^u}$, we deduce that ${f^u}$ takes the constant value ${a}$ on ${G':=\varphi_{(-\delta,\delta)}(G\cap W^s(w,\delta))}$. Furthermore, by putting together this fact with the initial observation that ${f^u}$ is constant along unstable manifolds ${W^u(.,\delta)}$, we obtain that ${f^u}$ takes the constant value ${a}$ on ${Sat^u(G',\delta)}$.

Note that, by assumption, ${G\cap W^s(w,\delta)}$ is a full volume subset of ${W^s(w,\delta)}$. Since ${\varphi_t}$ is a ${C^{\infty}}$-flow, it follows that ${G'}$ is a full volume subset of the ${n}$-dimensional smooth submanifold ${D=\varphi_{(-\delta,\delta)}(W^s(w,\delta))}$. Therefore, from the absolute continuity property of ${W^u}$ (cf. the second item of conclusion of Proposition 5), we conclude that ${Sat^u(G',\delta)}$ is a full volume subset of ${Sat^u(D,\delta):=N_{\delta}(w)}$. In particular, we have that ${f^u}$ takes the constant value ${a}$ on the full volume subset ${Sat^u(G',\delta)}$ of ${N_{\delta}(w)}$. Because ${f^u=B(f)}$ on ${G}$, we get that ${B(f)}$ takes the constant value ${a}$ on the full volume subset ${G\cap Sat^u(G',\delta)}$ of ${N_{\delta}(w)}$, i.e., ${B(f)|_{N_{\delta}(w)}}$ is almost everywhere constant. This completes the proof of the claim (a).

Remark 2 The reader is encouraged to interpret this argument in the light of Figure 2 in order to get a clear picture of the roles of the subsets ${G'}$, ${D}$ and ${N_{\delta}(w)}$.

We establish now the second claim (b) from the absolute continuity properties of the horospherical laminations and the local uniform transversality of the stable and unstable manifolds.

More concretely, from the absolute continuity property in the first item of the conclusion of Proposition 5, we have that the stable disk ${W^s(w,\delta)}$, resp. unstable disk ${W^u(w,\delta)}$, is almost everywhere tangent to the stable direction ${E^s}$, resp. unstable direction ${E^u}$, for almost every ${w\in\Lambda_2\cap V}$. Since the stable and unstable directions ${E^s}$ and ${E^u}$ are contained in the continuous families of cones ${\mathcal{C}^s}$ and ${\mathcal{C}^u}$ from Proposition 4, we have that ${W^s(w,\delta)}$, resp. ${W^u(w,\delta)}$, is everywhere tangent to ${\mathcal{C}^s}$, resp. ${\mathcal{C}^u}$ for almost every ${w\in\Lambda_2\cap V}$.

In particular, from the ${\varphi_t}$-invariance of the stable lamination ${W^s}$, we see that the ${n}$-dimensional disk ${D=D(w):=\varphi_{(-\delta,\delta)}(W^s(w,\delta))}$ is everywhere tangent to ${\mathcal{C}^s\oplus E^0}$ for almost every ${w\in\Lambda_2\cap V}$. Since the continuous conefields ${\mathcal{C}^s}$ and ${\mathcal{C}^u}$ meet only at the origin (cf. Proposition 4), that is, they are locally uniformly transverse, we conclude that there exists a neighborhood ${U_v}$ of ${v}$ such that

$\displaystyle W^u(w',\delta)\cap D(w)\neq\emptyset$

for almost any ${w, w'\in\Lambda_2\cap U_v}$. In other words, ${Sat^u(D(w)):=N_{\delta}(w)}$ intersects ${\Lambda_2\cap U_v}$ in a full volume subset. This completes the proof of claim (b).

This concludes our discussion of Hopf’s argument (namely, the derivation of claims (a) and (b)) for the ergodicity of ${\varphi_t}$.

Closing this post, let us say a few words about the mixing and Bernoulli properties in the statement of Theorem 1. In Burns-Masur-Wilkinson paper, these properties are deduced from general results of Katok saying that if a contact flow is non-uniformly hyperbolic and ergodic, then it is Bernoulli (and, in particular, mixing).

Nevertheless, as it was brought to my attention by B. Hasselblatt and Y. Coudène, the Hopf argument above can be slightly adapted in certain contexts to give mixing and/or mixing of all orders. For example, concerning the mixing property, Y. Coudène, B. Hasselblatt and S. Troubetzkoy showed (in Theorem 3.3) in this recent preprint here that if any ${L^2}$-function ${f}$ saturated by stable and unstable sets (in the sense that there is a full measure subset ${G}$ such that ${f(x)=f(y)}$ whenever ${x, y\in G}$ and ${y\in W^s(x)}$ or ${y\in W^u(x)}$) is almost everywhere constant, then the dynamical system is mixing. Also, they have a similar criterion for multiple mixing, and, furthermore, they discuss a couple of non-trivial examples of applications of their criteria.

In the context of Theorem 1, we can deduce the mixing property for ${\varphi_t}$ from the result of Coudène-Hasselblatt-Troubetzkoy. Indeed, the argument used in the proof of the claim (a) above (during the discussion of Hopf’s argument) also shows that any ${L^2}$-function saturated by stable and unstable sets (such as ${B(f)=f^s=f^u}$) is almost everywhere constant, so that Coudène-Hasselblatt-Troubetzkoy mixing criterion “à la Hopf” applies in this setting.

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