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This action has north-south dynamics and so, under iteration, points converge exponentially to $[T^\\phi_+]$.\n  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. 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