An introduction to ergodicity, cat maps and their quantization

Abstract

In this literary review, we introduce the topic of chaos and ergodic theory within the framework of both classical and quantum chaotic systems. We begin by delving into the definition of measures and integration with respect to a specific measure. Subsequently, we define transformations and discuss their properties of invariance and ergodicity. The tripling map and the quantum cat map are utilized as examples to illustrate these properties. Furthermore, for cat maps, we present a formal definition of quantization and develop the quantization process through the representation of the Heisenberg group. We discuss the relationship between classical and quantum dynamics through Egorov's theorem. Lastly, we discuss an application of these concepts: the Quantum Ergodicity theorem.