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

Theorem 1 (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 segmentinconnecting 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 satisfiesfor 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 boundfor every .

- (V) has
polynomially controlled injectivity radius, i.e., there are constants and such thatfor every (where denotes the injectivity radius at ).

- (VI) The
first derivative of the geodesic flowispolynomially 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) andergodic.

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, we proved in the previous post of this series that a geodesic flow 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 2Under the assumptions (II), (III) and (VI) in Theorem 1 above, the geodesic flow isnon-uniformly hyperbolic: more concretely, there exists a subset of full -measure such that the -invariant splitting

into the flow direction and the spaces and ofstable and unstable Jacobi fieldsalong have the property that

for all and .

Today, we want to exploit the non-uniform hyperbolicity of (and the assumptions (I) to (VI) above) in order to deduce the ergodicity of 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 : 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, 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 from Hopf’s argument.

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