Closed negatively curved manifolds have a Mather set of maximally stretched orbits that is nowhere dense unless the two metrics have proportional marked length spectra, in which case it is the whole unit tangent bundle.
The geometry of T eichm\" u ller space via geodesic currents
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Maximal stretch and Lipschitz maps on Riemannian manifolds of negative curvature
Closed negatively curved manifolds have a Mather set of maximally stretched orbits that is nowhere dense unless the two metrics have proportional marked length spectra, in which case it is the whole unit tangent bundle.