The Lattice Structure of Rotor-Router Model

In this paper, we study the rotor router model in the relation with the famous discrete dynamical system - Chip Firing Game. We consider the rotor router model as a discrete dynamical system defined on digraph and we use order theory to show that its state space started from any state is a lattice, which implies strong structural properties. The lattice structure of the state space of a dynamical system is of great interest since it implies convergence (and more) if the state space is finite. Moreover, we also attempt to define the class L(R) of lattices that are state space of a rotor router model, and compare it with the class of distributive lattices and the class of ULD lattices.