  The combinatorial theory of rotor-routers has connections with problems of
statistical mechanics, graph theory, chaos theory, and computer science. A
rotor-router network defines a deterministic walk on a digraph G in which a
particle walks from a source vertex until it reaches one of several target
vertices. Motivated by recent results due to Giacaglia et al., we study
rotor-router networks in which all non-target vertices have the same type. A
rotor type r is universal if every hitting sequence can be achieved by a
homogeneous rotor-router network consisting entirely of rotors of type r. We
give a conjecture that completely classifies universal rotor types. Then, this
problem is simplified by a theorem we call the Reduction Theorem that allows us
to consider only two-state rotors. A rotor-router network called the
compressor, because it tends to shorten rotor periods, is introduced along with
an associated algorithm that determines the universality of almost all rotors.
New rotor classes, including boppy rotors, balanced rotors, and BURD rotors,
are defined to study this algorithm rigorously. Using the compressor the
universality of new rotor classes is proved, and empirical computer results are
presented to support our conclusions. Prior to these results, less than 100 of
the roughly 260,000 possible two-state rotor types of length up to 17 were
known to be universal, while the compressor algorithm proves the universality
of all but 272 of these rotor types.
