  In this paper, we construct a geometrical compactification of the geodesic
flow of non-compact complete hyperbolic surfaces $\Sigma$ without cusps having
finitely generated fundamental group. We study the dynamical properties of the
compactified flow, for which we show the existence of attractive circles at
infinity. The geometric structure of ${\mathrm{T}}^1\Sigma$ for which this
compactification is realized is the pair of one-dimensional distributions
tangent to the stable and unstable horocyles of ${\mathrm{T}}^1\Sigma$. This is
a Kleinian path structure, that is a quotient of an open subset of the flag
space by a discrete subgroup $\Gamma$ of ${\mathrm{PGL}}_3(\mathbb{R})$. Our
study relies on a detailed description of the dynamics of
${\mathrm{PGL}}_3(\mathbb{R})$ on the flag space, and on the construction of an
explicit fundamental domain for the action of $\Gamma$ on its maximal open
subset of discontinuity in the flag space.

Dynamical Systems;Differential Geometry;Geometric Topology