Renormalization of quantum Anosov maps: Reduction to fixed boundary conditions

A renormalization scheme is introduced to study quantum Anosov maps (QAMs) on a torus for general boundary conditions (BCs), whose number (k) is alway s finite, it is shown that the quasienergy eigenvalue problem of a QAM for all k BCs is exactly equivalent to that of the renomalized QAM (with Planck 's constant h' = h/k) at some fixed BCs that can be of four types. The quan tum cat maps are, up to time reversal, fixed points of the renormalization transformation. Several results at fixed BCs, in particular the existence o f a complete basis of "crystalline" eigenstates in a classical limit, can t hen be derived and understood in a simple and transparent way in the genera l-BCs framework.