Asymmetric dynamics of outer automorphisms

We consider the action of an irreducible outer automorphism $\varphi$ on the closure of Culler–Vogtmann Outer space. This action has north-south dynamics and so, under iteration, points converge exponentially to $[T_{+}^{\varphi }]$. For each $N\ge 3$, we give a family of outer automorphisms $\phi_k \in $\text{Out}$($\mathbb{F}$_N)$suchthatas,$k$ goes to infinity, the rate of convergence of ${\varphi }_k$ goes to infinity while the rate of convergence of ${\varphi }_k^{-1}$ goes to one. Even if we only require the rate of convergence of ${\varphi }_k$ 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.