  This paper explores the orbit structure and homomesy (constant averages over
orbits) properties of certain actions of toggle groups on the collection of
independent sets of a path graph. In particular we prove a generalization of a
homomesy conjecture of Propp that for the action of a "Coxeter element" of
vertex toggles, the difference of indicator functions of symmetrically-located
vertices is 0-mesic. Then we use our analysis to show facts about orbit sizes
that are easy to conjecture but nontrivial to prove. Besides its intrinsic
interest, this particular combinatorial dynamical system is valuable in
providing an interesting example of (a) homomesy in a context where large orbit
sizes make a cyclic sieving phenomenon unlikely to exist, (b) the use of
Coxeter theory to greatly generalize the set of actions for which results hold,
and (c) the usefulness of Striker's notion of generalized toggle groups.
