Compact composition operators on BMOA

We characterize the compact composition operators on BMOA, the space consis ting of those holomorphic functions on the open unit disk U that are Poisso n integrals of functions on partial derivative U, that have bounded mean os cillation. We then use our characterization to show that compactness of a c omposition operator on BMOA implies its compactness on the Hardy spaces (a simple example shows the converse does not hold). We also explore how compa ctness of the composition operator C phi : BMOA --> BMOA relates to the sha pe of phi(U) near partial derivative U, introducing the notion of mean orde r of contact. Finally, we discuss the relationships among compactness condi tions for composition operators on BMOA, VMOA, and the big and little Bloch spaces.