Exact Renormalization Scheme for Quantum Anosov Maps

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An exact renormalization scheme is introduced for quantum Anosov maps (QAMs) on a torus for general boundary conditions (BCs), whose number is always finite. Given a QAM U with k BCs and Planck's constant ℏ =2π/p (p integer), its nth renormalization iterate U⁽ⁿ⁾= Rⁿ(U) is associated with k BCs for all n and with a Planck's constant ℏ ⁽ⁿ⁾=ℏ /kⁿ. It is shown that the quasienergy eigenvalue problem for U⁽ⁿ⁾ for all k BCs is equivalent to that for U⁽ⁿ⁺¹⁾ at some fixed BCs, corresponding, for n>0, to either strict periodicity for kp even or antiperiodicity for kp odd. The quantum cat maps are, in general, fixed points of either R or R². The Hannay-Berry results turn out then to be significant also for general BCs.