  We study the open system dynamics and steady states of two dimensional
Floquet topological insulators: systems in which a topological Floquet-Bloch
spectrum is induced by an external periodic drive. We solve for the bulk and
edge state carrier distributions, taking into account energy and momentum
relaxation through radiative recombination and electron-phonon interactions, as
well as coupling to an external lead. We show that the resulting steady state
resembles a topological insulator in the Floquet basis. The particle
distribution in the Floquet edge modes exhibits a sharp feature akin to the
Fermi level in equilibrium systems, while the bulk hosts a small density of
excitations. We discuss two-terminal transport and describe the regimes where
edge-state transport can be observed. Our results show that signatures of the
non-trivial topology persist in the non-equilibrium steady state.
