Bounding the gap between a free group (outer) automorphism and its inverse

For any finitely generated group $G$, two complexity functions $\alpha_G$ and $\beta_G$ are defined to measure the maximal possible gap between the norm of an automorphism (respectively outer automorphism) of $G$ and the norm of its inverse. Restricting attention to free groups $F_r$, the exact asymptotic behaviour of $\alpha_2$ and $\beta_2$ is computed. For rank $r\geqslant 3$, polynomial lower bounds are provided for $\alpha_r$ and $\beta_r$, and the existence of a polynomial upper bound is proved for $\beta_r$.Comment: 24 pages; To appear in Collectanea Mathematic