Hyperbolic automorphisms of free groups

We prove that an automorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ {\not o} : F \to F $\end{document} of a finitely generated free group F is hyperbolic in the sense of Gromov if it has no nontrivial periodic conjugacy classes. This result was previously claimed (but not proved) in [BF1].