The geodesic flow
Given any Riemannian manifold M, we may define a geodesic flow on the unit tangent bundle which sends a point (x, v) to the point , where
- is the point distance t from x along the geodesic ray emanating from x in the direction of v, and
- is the parallel transport of v along the same ray
(it’s a mouthful, isn’t it? It’s really simpler than all those words make it seem.) Note, at each point, we remember not just where we are—the point —, but also where we’re going—the direction vector ; if we were to forget this second piece of information, we would become a little unmoored: here we are … where should we go next?
When M is a closed (i.e. compact, no boundary) hyperbolic surface, or more generally closed with strictly negative curvature, this geodesic flow is ergodic, i.e. any subset of or M invariant under the flow has either zero measure, or full measure. Here the measure on our Riemannian manifold is the pushforward of the Lebesgue measure through the coordinate charts.
Since linear combinations of step functions are dense in the space of bounded measurable functions, we may equivalently define ergodicity as: any measurable function invariant under the flow is a.e. constant.
(Side note: with more assumptions on the curvature we may relax the compactness assumption to a finite volume assumption)
The Hopf argument (for closed hyperbolic manifolds)
This is essentially due to the exponential divergence of geodesics in negative curvature , and the splitting of the tangent spaces into stable, tangent (flowline), and unstable distributions; these give rise to three maximally transverse foliations, the stable foliation , the unstable foliation , and the foliation by flowlines .
The flow is exponentially contracting in the forward time direction on the leaves of the stable foliation , and on which the flow is exponentially contracting in the reverse time direction the leaves of the unstable foliation . In other words, the flow is Anosov.
We may describe these foliations explicitly in the case of constant negative curvature—if we take to be the geodesic tangent to ,
- (v) is (the quotient image of) the unit normal bundle to the horosphere through tangent to the forward endpoint of in . “forward” here being taken with reference to how v is pointing along ;
- is (the quotient image of) the unit normal bundle to the horosphere through tangent to the backward endpoint of in ;
Suppose f is a -invariant function; by replacing f with min(f, C) if needed, WMA f is bounded. Since continuous functions are dense in the set of measurable functions on M, we may approximate f in by bounded continuous functions .
By the Birkhoff ergodic theorem, forward time averages [w.r.t. ] exist for .
By an argument involving the -invariance of f and the triangle inequality, f is well-approximated (in ) by the forward time averages of .
The forward time averages of are constant a.e., since by invariance these averages are already constant a.e. on (each of) the leaves of , and they are also constant a.e. on (each of the) unstable and stable leaves, by uniform continuity of .
(See also Proposition 2.6 in Brin’s exposition)
To conclude that time averages, and hence our original arbitrary integrable function, are constant a.e. on M, we (would like to!) use Fubini’s theorem: locally near each , the set of (x, v) along each of the foliation directions at which the time averages are equal to those at has full measure, by the previous Step.
By Fubini’s theorem applied to the three foliation directions, we (would) conclude that the set of nearby (x, v) at which the time averages are equal to those at has full measure. Hence the time averages are locally constant, and since is connected we are done.
But! (Also more generally, for K < 0)
The problem is that while our stable and unstable leaves are differentiable, the foliations need not be—i.e. the leaves may not vary smoothly in their parameter space.
To justify the use of a Fubini-type argument one instead shows that that these foliations are absolutely continuous.
The proof then immediately generalizes to all compact manifolds with (not necessarily constant) negative sectional curvature. For more general negatively-curved nanifolds, the stable and unstable foliations and may still be described in terms of unit normal bundles over horospheres, where horospheres are now described, more generally, as level sets of Busemann functions.
The proof of absolute continuity of the foliations proceeds as follows
- Showing that the stable and unstable distributions and (also the “central un/stable” or “weak un/stable” distributions, i.e. and ) (of any Anosov flow) are Hölder continuous—i.e. given , the Hausdorff distance in between the stable subspace and the stable subspace is .
Roughly speaking, this is true because any complementary subspace to will become exponentially close to under the repeated action of the geodesic flow, by the same mechanism that makes power iteration tick; and the distance function on M is Lipschitz. Analyzing the situation more carefully, and applying a bunch of simplifying tricks such as the adjusted metric described in Brin’s section 4.3, yields the desired Hölder continuity.
- Using this, together with the description of horospheres as limits of sequences of spheres with radii increasing to , to establish that between any pair of transversals for the un/stable foliation, we have a homeomorphism which is with bounded Jacobians, and hence absolutely continuous.
Very slightly less vaguely, Hölder continuity of , together with the power iteration argument as above, implies tangents to transversals to the stable foliation become exponentially close; given regularity of the Riemannian metric, this implies the Jacobians of the iterated geodesic flow on these transversals become exponentially close. By a chain rule argument and another application of the power iteration argument, this implies that the Jacobians of the map between transversals are bounded.
This condition on the foliations is known as transversal absolute continuity, and implies, by a general measure theoretic argument (see section 3 of Brin’s article), absolute continuity of the foliations.
- Note that this last step, at least as presented in Brin, appears to require the use of pinched negative curvature.
Eberhard Hopf, “Ergodic theory and the geodesic flow on surfaces of constant negative curvature.” Bull. Amer. Math. Soc. 77 (1971), no. 6, 863–877.
Yves Coudene, “The Hopf argument.”