A CANONICAL QUANTIZATION OF THE BAKERS MAP

We present here a canonical quantization for the baker's map. The meth od we use is quite different from that used in Balazs and Voros and Sa raceno. We first construct a natural ''baker covering map'' on the pla ne R-2. We then use as the quantum algebra of observables the subalgeb ra of operators on L-2(R) generated by {exp(2 pi i (x) over cap), exp( 2 pi i (p) over cap)}. We construct a unitary propagator such that as (h) over bar --> 0 the classical dynamics is returned. For Planck's co nstant h = 1/N, we show that the dynamics can be reduced to the dynami cs on an N-dimensional Hilbert space, and the unitary N x N matrix pro pagator is the same as given by Balazs and Voros, except for a small c orrection of order h. This correction is shown to preserve the classic al symmetry x --> 1 - x and p --> 1 - p in the quantum dynamics for pe riodic boundary conditions. (C) 1998 Academic Press.