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We show that most compact semi-simple Lie groups carry many left invariant metrics with
positive topological entropy. We also show that many homogeneous spaces admit collective
Riemannian metrics arbitrarily close to the bi-invariant metric and whose geodesic flow
has positive topological entropy. Other properties of collective geodesic flows are also
discussed.
A. J. Montesinos has shown that a geodesic correspondence between two complete Riemannian manifolds with transitive topological geodesic flow is a homothety. In this paper we prove a similar result for a conformal geodesic correspondence between two singular flat surfaces with conical singularities and negative concentrated curvature.
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