  We give a complete characterization of the spectrum of composition operators,
induced by an automorphism of the open unit disk, acting on a family of Banach
spaces of analytic functions that includes the Bloch space and BMOA. We show
that for parabolic and hyperbolic automorphisms, the spectrum is the unit
circle. For the case of elliptic automorphisms, the spectrum is either the unit
circle or a finite cyclic subgroup of the unit circle.
