  In this paper we study topological aspects of the dynamics of the foliated
horocycle flow on flat projective bundles over hyperbolic surfaces and we
derive ergodic consequences. If $\rho : \Gamma \to {\rm PSL}(n+1,\mathbb{R})$
is a representation of a non-elementary Fuchsian group $\Gamma$, the unit
tangent bundle $Y$ associated to the flat projective bundle defined by $\rho$
admits a natural action of the affine group $B$ obtained by combining the
foliated geodesic and horocycle flows. If the image $\rho(\Gamma)$ satisfies
Conze-Guivarc'h conditions, namely strong irreducibility and proximality, the
dynamics of the $B$-action is captured by the proximal dynamics of
$\rho(\Gamma)$ on $\mathbb{R}{\rm P}^n$ (Theorem A). In fact, the dynamics of
the foliated horocycle flow on the unique $B$-minimal subset of $Y$ can be
described in terms of dynamics of the horocycle flow on the non-wandering set
in the unit tangent bundle $X$ of the surface $S= \Gamma \backslash \mathbb{H}$
(Theorem B). Assuming the existence of a continuous limit map, we prove that
the $B$-minimal set is an attractor for the foliated horocycle flow restricted
to the proximal part of the non-wandering set in $Y$ (Theorem C). As a
corollary, we deduce that the restricted flow admits a unique conservative
ergodic $U$-invariant Radon measure (defined up to a multiplicative constant)
if and only if $\Gamma$ is convex-cocompact. For example, the foliated
horocycle flow on the projective bundle defined by the Cannon-Thurston map is
uniquely ergodic.
