Homological shadows of attracting laminations

Abstract

Given a free group $F_n$, a fully irreducible automorphism $f\in \mathrm{Aut}(F_n)$, and a generic element $x\in F_n$, the elements $f^k(x)$ converge in the appropriate sense to an object called an attracting lamination of $f$. When the action of $f$ on $\frac{F_n}{[F_n,F_n]}$ has finite order, we introduce a homological version of this convergence, in which the attracting object is a convex polytope with rational vertices, together with a measure supported at a point with algebraic coordinates.