Exact Renormalization Scheme for Quantum Anosov Maps

Abstract

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 $\hslash =2\pi /p$ ($p$ integer), its $n$th renormalization iterate ${\hat{U}}^{(n)}={\mathcal{R}}^n(\hat{U})$ is associated with $k$ BCs for all $n$ and with a Planck's constant ${\hslash }^{(n)}=\hslash /k^n$. 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 $\mathcal{R}$ or ${\mathcal{R}}^2$. The Hannay-Berry results turn out then to be significant also for general BCs.

12 pages, REVTEX, no figures