  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.
