The Lattice Structure of Rotor-Router Model

Abstract

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 $L(\mathcal R)$ 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.