Quantization of a class of piecewise affine transformations on the torus

We present a unified framework for the quantization of a family of discrete dynamical systems of varying degrees of ''chaoticity.'' The systems to be quantized are piecewise affine maps on the two-torus, viewed as phase space, and include the automorphisms, translations and skew translations. We then treat some discontinuous transformations such as the Baker map and the sawtooth-like maps. Our approach extends some ideas from geometric quantization and it is both conceptually and calculationally simple.