[joint work with P. Silva and M. Ladra] Let $G$ be a group and $X$ a finite set of generators. In this talk, we will introduce a function $f(n)$ whose asymptotic behavior measures how difficult it is to invert automorphisms of $G$. More precisely, $f(n)$ is the maximum norm of the inverses of all those automorphisms of $G$ whose norm is less than or equal to $n$. After analyzing some general properties of this function, we will restrict our attention to the free case. For all such groups we will show that $f(n)$ is polynomial. More specifically, it will be proven that, in rank r, $f(n)$ is bounded below by a polynomial of degree $r$; and above by a quadratic polynomial in the special case of rank 2. We obtain these results essentially by abelianizing and playing with matrices and eigenvalues, and by arguing directly in the free group in the very special case of rank 2. General upper bounds for $f(n)$ are more complicated to obtain: the main result in the talk is an upper bound in arbitrary rank, by a polinomial of big enough degree $M(r)$, depending only on the ambient rank $r$. The main ingredient in the proof of this last inequality is a recent result by AlgomKfir-Bestvina, about the asimmetry of the metric in the outer space. The rest of the talk will be devoted to recent developements regarding fixed points of automorphisms and endomorphisms of free groups.