QUANTIZATION OF A CLASS OF PIECEWISE AFFINE TRANSFORMATIONS ON THE TORUS

TL;DR

This paper develops a unified, simple framework for quantizing various piecewise affine dynamical systems on the two-torus, including chaotic and discontinuous maps, extending geometric quantization methods.

Contribution

It introduces a novel, unified approach to quantize a broad class of piecewise affine maps on the torus, including chaotic and discontinuous systems.

Findings

Framework applicable to automorphisms, translations, skew translations

Extension to discontinuous maps like Baker and sawtooth maps

Approach is conceptually simple and computationally effective

Abstract

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.