Exact Renormalization Scheme for Quantum Anosov Maps

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 $\hat{U}$ with $k$ BCs and Planck's constant $\hbar =2\pi /p$ ($p$ integer), its $n$th renormalization iterate $\hat{U}^{(n)}={\cal R}^{n}(\hat{U})$ is associated with $k$ BCs for all $n$ and with a Planck's constant $\hbar ^{(n)}=\hbar /k^{n}$. It is shown that the quasienergy eigenvalue problem for $\hat{U}^{(n)}$ for {\em all} $k$ BCs is equivalent to that for $\hat{U}^{(n+1)}$ at some {\em fixed} BCs, corresponding, for $n>0$, to either strict {\em periodicity} for $kp$ even or {\em antiperiodicity} for $kp$ odd. The quantum cat maps are, in general, fixed points of either ${\cal R}$ or ${\cal R}^{2}$. The Hannay-Berry results turn out then to be significant also for general BCs.