The geometry of deviations from the asymptotic cycle

Abstract

Let w be a word in a free group, and let f be an automorphism. When theabelianization of f is the identity, it is known that lifts of the words f^k(w) to the cover corresponding to the commutator subgroup converge, in a scaled sense, to a convex polytope with rational vertices. I show that when the abelianization of f is not the identity, the lifts of the words f^k(w) to the same cover still have a notion of convergence to a geometric structure. This can be used to describe the asymptotic behavior of foliations of pseudo-Anosov homeomorphisms in the homology of punctured flat surfaces. The behavior I exhibit differs substan- tially from the known behavior of almost every other foliation. I demonstrate this difference visually with experiments that further suggest the presence of attracting geometric objects for all foliations.