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
$M=\mathbb H^2/\Gamma$
of constant negative curvature has a non-compact non-trivial minimal set
if and only if the Fuchsian group $\Gamma$ 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
$\operatorname{SL}(2,\mathbb R)$ with all orbits discrete and dense in
$\mathbb R^2$ is constructed.
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