Let G be a torsion-free hyperbolic group and α an automorphism of G. We show that there exists a canonical collection of subgroups that are polynomially growing under α, and that the mapping torus of G by α is hyperbolic relative to the suspensions of the maximal polynomially growing subgroups under α. As a consequence, we obtain a dichotomy for growth: given an automorphism of a torsion-free hyperbolic group, the conjugacy class of an element either grows polynomially under the automorphism, or at least exponentially.