Horocycle flow on flat projective bundles: topological remarks and applications

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 sphere bundle defined by the Cannon-Thurston map is uniquely ergodic.