The Hopf argument
Yves Coudène
Journal of Modern Dynamics 2007, 1(1): 147-153 doi: 10.3934/jmd.2007.1.147
Let $T$ be a measure-preserving transformation of a metric space $X$. Assume $T$ is conservative and $X$ can be covered by a countable family of open sets, each of finite measure. Then any eigenfunction is invariant with respect to the stable foliation of $T$.
keywords: ergodicity. Hopf argument
Counterexamples in non-positive curvature
Yves Coudène Barbara Schapira
Discrete & Continuous Dynamical Systems - A 2011, 30(4): 1095-1106 doi: 10.3934/dcds.2011.30.1095
We give examples of rank one compact surfaces on which there exist recurrent geodesics that cannot be shadowed by periodic geodesics. We build rank one compact surfaces such that ergodic measures on the unit tangent bundle of the surface are not dense in the set of probability measures invariant by the geodesic flow. Finally, we give examples of complete rank one surfaces for which the non wandering set of the geodesic flow is connected, the periodic orbits are dense in that set, yet the geodesic flow is not transitive in restriction to its non wandering set.
keywords: Dynamical systems geodesic flow negative curvature.

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