In the first post of this series, we planned to discuss in the third and fourth posts the proof of the following ergodicity criterion for geodesic flows in incomplete negatively curved manifolds of Burns-Masur-Wilkinson:

Theorem 1 (Burns-Masur-Wilkinson)Let $latex {N}&fg=000000$ be the quotient $latex {N=M/Gamma}&fg=000000$ of a contractible, negatively curved, possibly incomplete, Riemannian manifold $latex {M}&fg=000000$ by a subgroup $latex {Gamma}&fg=000000$ of isometries of $latex {M}&fg=000000$ acting freely and properly discontinuously. Denote by $latex {overline{N}}&fg=000000$ the metric completion of $latex {N}&fg=000000$ and $latex {partial N:=overline{N}-N}&fg=000000$ the boundary of $latex {N}&fg=000000$.Suppose that:

- (I) the universal cover $latex {M}&fg=000000$ of $latex {N}&fg=000000$ is
geodesically convex, i.e., for every $latex {p,qin M}&fg=000000$, there exists an unique geodesic segmentin$latex {M}&fg=000000$ connecting $latex {p}&fg=000000$ and $latex {q}&fg=000000$.- (II) the metric completion $latex {overline{N}}&fg=000000$ of $latex {(N,d)}&fg=000000$ is
compact.- (III) the boundary $latex {partial N}&fg=000000$ is
volumetrically cusplike, i.e., for…

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