Equivalence of Deterministic walks on regular lattices on the plane

We consider deterministic walks on square, triangular and hexagonal two dimensional lattices. In each case, there is a scatterer at every site that can be in one of two states that force the walker to turn either to his/her immediate right or left. After the walker is scattered, the scatterer changes state. A lattice with an arrangement of scatterers is an environment. We show that there are only two environments for which the scattering rules are injective, mirrors or rotators, on the three lattices. On hexagonal lattices, B. Z. Webb and E. G. D. Cohen, proved that given an initial position and velocity of the walker and an environment of one type of scatterers, mirrrors or rotators, there is an environment of the other type such that the walks on both environments are equivalent, meaning they visit the same sites at the same time steps. We prove the equivalence of walks on square and triangular lattices and include a proof of the equivalence of walks on hexagonal lattices. The proofs are based both on the geometry of the lattice and the structure of the scattering rule.