Introduction

Fixed point theorems are usually purely topological in nature, and do not usually have any measure theoretic hypotheses. However, there are three surfaces where the assumption that a homeomorphism is area preserving, by itself or with additional assumptions, implies the existence of a fixed point: the open square, the torus, and the annulus. The reason only 2-dimensional manifolds are covered is that all these results follow from a purely topological fixed point theorem of Brouwer for homeomorphisms of the plane, known as the ‘Plane Translation Theorem’. This theorem says that if an orientation preserving homeomorphism of the plane has no fixed point then it is ‘like a translation’. This phrase can be made precise in various ways, but it will be sufficient for our purposes here to take it to mean ‘has no periodic points’.

Since the issue of fixed points is not a main concern of this book, we will not attempt to give the strongest forms of theorems, but merely show how results obtained earlier in the book can give simple demonstrations of the existence of fixed points. References to the stronger results of Franks and Flucher will be given.

The organization of this chapter is as follows. In Section 5.2 we state a special case of Brouwer's Plane Translation Theorem due to Andrea [32]. We apply this in Section 5.3 to prove a result of Montgomery [86] that any orientation preserving, area preserving homeomorphism of the open square has a fixed point.