The introduction of this article is extremely well written, and it gives a clear overview of the methods used, the technical difficulties that had to be overcome and the historical development of the subject.
The authors prove the ergodicity of the geodesic flow on \(\mathfrak{M}(S)\), where \(S\) is a compact surface of genus \(g\) with \(n \geq 0\) punctures and \(\mathfrak{M}(S)\) denotes the moduli space of conformal structures (up to equivalence) on \(S\). The space \(\mathfrak{M}(S)\) has the structure of a complex orbifold of dimension \(3g - 3 + n\), and the universal cover is the Teichmüller space Teich(\(S\)). On Teich(\(S\)) there is a WP metric (Weil-Petersson) and a symplectic form that descend to \(\mathfrak{M}(S)\). \(\mathfrak{M}(S)\) has finite volume with respect to the volume element defined by the symplectic form. The sectional curvatures of the WP metric are negative, but the WP metric is incomplete. This means that the methods used for studying complete, negatively curved manifolds are not immediately available here, and must be recovered with special methods and considerable effort. The authors actually prove that the geodesic flow is Bernoulli, which implies mixing of all orders. Earlier work of Brock, Masur and Minsky had proved the existence of dense orbits of the geodesic flow and the denseness of the vectors with periodic orbits. These authors also proved that the topological entropy is infinite, in contrast to the finiteness of the measure theoretic entropy proved in this article.
Until roughly 1940 the results about the geodesic flow on a compact, negatively curved manifold \(M\) were confined to manifolds of dimension two and constant negative Gaussian curvature. Ergodicity in this setting was known. The methods used were algebraic in origin. Around 1940 E. Hopf discovered an ingenious way to study compact surfaces of variable negative curvature by using the interplay of the stable and unstable foliations of the geodesic flow. In fact, these methods proved ergodicity for the geodesic flow of manifolds of arbitrary dimension and constant negative sectional curvature, but they required the stable and unstable foliations to be \(C^{1}\). The \(C^{1}\) condition is not satisfied for variable negative sectional curvature in dimensions \(\geq 3\). Hopf’s simple and elegant idea for proving ergodicity has come to be called the Hopf Argument, and it has served as a blueprint for proving ergodicity in related settings. However, typically one must overcome significant technical difficulties not present in Hopf’s setting before one can apply the Hopf Argument. In particular, one must prove the almost everywhere existence of stable and unstable manifolds and the absolute continuity of the corresponding foliations. For the geodesic flow of a compact manifold with variable negative sectional curvature this followed from the theory of hyperbolic dynamical systems developed in the 60s by Anosov and Sinai. However, the absolute continuity of these foliations is typically unknown (or even false) for nonuniformly hyperbolic flows, including the geodesic flow of a compact surface of nonpositive Gaussian curvature.
In the present setting, the geodesic flow on \(\mathfrak{M}(S)\), the authors prove the local existence and smoothness of stable and unstable manifolds with the help of existence criteria established by Katok and Strelcyn. They find bounds on the norm of the first and second derivatives of the geodesic flow in a suitable domain to show that these criteria are satisfied. The importance of such bounds had been noted earlier by Pollicott and Weiss. Establishing these bounds on the norm uses earlier results of Wolpert and McMullen. Finally, the authors use the negative curvature and geodesic convexity of the WP metric to establish the almost everywhere existence of the stable and unstable manifolds. The Hopf Argument can now be applied with small modifications.