Asymmetric dynamics of outer automorphisms

TL;DR

This paper studies the asymmetric convergence rates of outer automorphisms on Outer space, providing explicit examples where the automorphism and its inverse have vastly different convergence behaviors, especially for rank at least 3.

Contribution

It constructs explicit families of outer automorphisms demonstrating asymmetric convergence rates, revealing new insights into their dynamics and properties.

Findings

Convergence rate of automorphisms can be arbitrarily fast compared to their inverses for rank ≥ 3.

No uniform bound exists on the distance between axes of automorphisms and their inverses.

Asymmetry in dynamics depends on the rank of the free group.

Abstract

We consider the action of an irreducible outer automorphism $\phi$ on the closure of Culler--Vogtmann Outer space. This action has north-south dynamics and so, under iteration, points converge exponentially to $[T^\phi_+]$. For each $N \geq 3$, we give a family of outer automorphisms $\phi_k \in \textrm{Out}(\mathbb{F}_N)$ such that as, $k$ goes to infinity, the rate of convergence of $\phi_k$ goes to infinity while the rate of convergence of $\phi_k^{-1}$ goes to one. Even if we only require the rate of convergence of $\phi_k$ to remain bounded away from one, no such family can be constructed when $N < 3$. This family also provides an explicit example of a property described by Handel and Mosher: that there is no uniform upper bound on the distance between the axes of an automorphism and its inverse.