Abstract
In the first part of the paper the following conjecture stated by Dal'bo and Starkov is proved: the geodesic flow on a surface of constant negative curvature has a non-compact non-trivial minimal set if and only if the Fuchsian group is infinitely generated or contains a parabolic element. In the second part interesting examples of horocycle flows are constructed: 1) a flow whose restriction to the non-wandering set has no minimal subsets, and 2) a flow without minimal sets. In addition, an example of an infinitely generated discrete subgroup of with all orbits discrete and dense in is constructed.
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