  We demonstrate the directed transport of underdamped particles in two
dimensional lattices of arbitrary geometry driven by an unbiased ac-driving
force. The direction of transport can be controlled via the lattice geometry as
well as the strength and orientation of the oscillating drive. The breaking of
the spatial inversion symmetry, which is necessary for the emergence of
directed transport, is achieved solely due to the structure and geometry of the
lattice. The most important criterion determining the transport direction is
shown to be the ballistic attractors underlying the phase space of our weakly
dissipative non-linear dynamical system. This allows the prediction of
transport direction even for setups like driven oblique lattices where the
standard symmetry arguments of transport control fail. Our results can be
experimentally realized using holographic optical lattice based setups with
colloids or cold atoms.
