Introduction

The central theme of this book is that the mathematical notion of a linked twist map, and attendant dynamical consequences, is naturally present in a variety of different mixing situations. In this chapter we will define what we mean by a linked twist map, and then give a general idea of why they capture the essence of ‘good mixing’. To do this we will first describe the notion of a linked twist map as first studied in the mathematical literature. This setting may at first appear to have little to do with the types of situations arising in fluid mechanics, but we will argue the contrary later. However, this more mathematically ideal setting allows one to rigorously prove strong mixing properties in a rather direct fashion that would likely be impossible for the types of maps arising in typical fluid mechanical situations. We will then consider a variety of mixers and mixing situations and show how the linked twist map structure naturally arises.