Invariant laminations for irreducible automorphisms of free groups

TL;DR

This paper explores the structure of invariant laminations associated with atoroidal iwip automorphisms of free groups, revealing a finite procedure to derive the dual lamination from the backward limit current, analogous to pseudo-Anosov homeomorphisms.

Contribution

It introduces a finite procedure to obtain the dual lamination from the backward limit current and provides new characterizations and a structure theorem for this lamination.

Findings

Dual lamination is the diagonal closure of the support of the backward limit current.

A finite procedure analogous to adding diagonal leaves constructs the dual lamination.

New characterizations and a structure theorem relate the dual lamination to stable laminations.

Abstract

For every atoroidal iwip automorphism $\phi$ of $F_N$ (i.e. the analogue of a pseudo-Anosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree $T_+(\phi)$ is obtained as "diagonal closure" of the support of the backward limit current $\mu_-(\phi)$. This diagonal closure is obtained through a finite procedure in analogy to adding diagonal leaves from the complementary components to the stable lamination of a pseudo-Anosov homeomorphism. We also give several new characterizations as well as a structure theorem for the dual lamination of $T_+(\phi)$, in terms of Bestvina-Feighn-Handel's "stable lamination" associated to $\phi$.