Asymmetric dynamics of outer automorphisms

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.