Introduction

The main goal of the mathematical sections of this book is to show that linked twist maps have the Bernoulli property on all of their domain (except for possibly a set of measure zero). Before discussing the theory that will be necessary to attack this problem, we start with an easier, preliminary result. Namely, we give Devaney's proof of a theorem that a linked twist map has a Smale horseshoe (Devaney (1978)). This is a somewhat ambiguous, albeit commonly used statement in the literature. The Smale horseshoe map is a homeomorphism (it need not be area preserving) having the property that it has an invariant set on which the map is topologically conjugate to the Bernoulli shift, i.e., it has the Bernoulli property on an invariant set. A slight confusion may arise since occasionally the invariant set itself is referred to as the horseshoe. Smale horseshoe (or just “horseshoe”) maps are ubiquitous in the sense that they can always be constructed near transverse homoclinic points. This is the content of the Smale–Birkhoff homoclinic theorem. All of this is described in detail, and from an elementary point of view, in Wiggins (2003).