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.

Introduction

The Lusin Theorem (or rather its consequence Corollary 6.3) in the previous chapter provides us with a method of constructing volume preserving homeomorphisms with desired measure theoretic properties. This method reduces the problem to approximating a volume preserving homeomorphism uniformly by a volume preserving automorphism (not necessarily continuous) with the desired measure theoretic property. In the next chapter we will give a very general application of this method, but here we use it simply to demonstrate the existence (and typicality) of ergodic homeomorphisms of the cube. (We recall that an automorphism of a finite measure space is said to be ergodic if its only invariant sets are of measure zero or full measure.) Again, this is an optional chapter, in that a stronger result (Theorem 8.2) will be proved independently in the next chapter.

However, the proof we present here, that ergodicity is typical among volume preserving homeomorphisms of the cube, is a very clear illustration of the method of approximation by discontinuous automorphisms. Given Corollary 6.3 of the previous chapter, we are required only to approximate an arbitrary homeomorphism in M[In, λ] by an ergodic (generally discontinuous) automorphism in G[In, λ], in the uniform topology.

Theorem 7.1The ergodic homeomorphisms form a dense Gδ subset of the volume preserving homeomorphisms of In, in the uniform topology.

Proof Let G[In, λ] denote the space of all volume (λ) preserving bimeasurable bijections of the unit cube, endowed with the weak topology.

Introduction

In this chapter we determine necessary and sufficient conditions for a measure preserving homeomorphism h of a sigma compact manifold X to be the limit of ergodic homeomorphisms, in the compact-open topology. We have already shown in Prasad's Theorem 12.4 that such an approximation is always possible (for any h) when X is Euclidean space Rn with Lebesgue measure. In Examples 13.1 and 13.2 we showed on the contrary that when h is the unit translation on either the Manhattan manifold or the strip manifold, such an approximation is not possible. The obstruction to an ergodic approximation for these systems was explained in terms of the ends E of the manifold X in Chapter 14: The unit translation on the Manhattan manifold induces a homeomorphism on the ends which is compressible, and hence ergodic approximation is precluded by Lemma 14.15; the unit translation on the strip manifold is incompressible, but since it induces a nonzero charge on the ends, an ergodic approximation is ruled out by Theorem 14.23. Conditions on the ends will be used in this chapter to obtain positive results on ergodic approximation. The results in this chapter come from [20] and [22].

We will obtain complete answers to a number of simple questions regarding ergodic approximation in M[X, μ] with respect to the compactopen topology.

Transitive Homeomorphisms

A mapping h of a topological space is called transitive if for any open sets U and V, there is a positive integer k with hkU ∩ V ≠ ø. For manifolds (actually, for any complete, separable metric space without isolated points) this is equivalent to the existence of a dense orbit under h. For some spaces, it is easy to exhibit transitive homeomorphisms (e.g., irrational rotations of the circle), while for other spaces where they exist more subtlety is required. Besicovitch [40] first defined such a transformation for the plane in 1937, partially answering a question of Ulam (see Chapter 12). In the same year, Oxtoby [89] used the Baire Category Theorem to demonstrate that in fact transitivity is typical for volume preserving homeomorphisms of the cube. At that time, the existence of such transformations had not been established. Recently Xu ([108, 109]) has shown how Besicovitch's transitive homeomorphism of the plane can be used to construct an explicit transitive homeomorphism of the closed unit square and Cairns, Jessup and Nicolau [48] have given examples on the 2-sphere and more generally on quotients of tori.

In this chapter we present a simplification of Oxtoby's original proof based on the combinatorial techniques of Chapter 3, and then we extend this method to show the existence of spatially periodic transitive homeomorphisms of Rn, or equivalently, rotationless homeomorphisms of the torus with transitive lifts.

Noncompact Manifolds

Up to now, we have considered dynamics on compact manifolds with finite measures. In this last part of the book we widen our analysis to include noncompact manifolds and consequently infinite measures.

• Topologically, the analysis extends to cover sigma compact manifolds X – manifolds which can be represented as a countable union of compact sets. In fact (see Section 14.6), they can be represented as a countable union of compact manifolds. As in the compact case, we allow a manifold boundary, which we denote by ∂X. For noncompact manifolds, the notion of an end (roughly, a way of going to infinity) will turn out to be of great importance. This notion will be introduced informally in Chapter 13, and then more formally in Chapter 14.

• Measure theoretically, the manifold X will be endowed with a fixed OU measure μ which can be finite or infinite, but in any case the definition of an OU measure ensures it is sigma finite. This means the space X can be written as a countable union of sets of finite μ-measure. Mainly we will be interested in the case where the OU measure μ is infinite, as the finite measure case resembles the theory developed earlier for compact manifolds. The relation between the ends of the manifold X and the measure μ will be important for the theory we will develop. Some ends will have infinite measure, and those ends of infinite measure will be significant in the theory.

Introduction

Up to this point we have shown that ergodicity (as well as other properties) is generic for homeomorphisms of any compact manifold and for Euclidean space Rn. The reader may naturally expect that we will continue in this fashion and show that generic ergodicity holds for any noncompact manifold. The purpose of this chapter is to show that this is not the case by presenting two measured manifolds (X, μ) for which ergodicity is not generic in the space M[X, μ]. After presenting these two examples, we will use them to motivate the notion of an end of a noncompact space. We will give an informal discussion of how the behavior of a homeomorphism h ∈ M[X, μ] with respect to the ends of the manifold X can prevent it, or any homeomorphism close to it (in the compact-open topology), from being ergodic. The two types of behavior found in the counterexamples given in this chapter (namely compressibility and nonzero charge) will have to be excluded, by hypothesis, in the following chapters. In those chapters we will give positive results on the typicality of ergodicity or other dynamical properties in certain closed subspaces of M[X, μ] for general noncompact manifolds X.

Two Examples

In both of the examples of measured manifolds (X, μ) given below, the manifold X is a subset of the plane.

Introduction

The study of typical properties of volume preserving homeomorphisms of noncompact manifolds was initiated by Prasad [96], with a proof that ergodicity is generic when the manifold is Euclidean n-space Rn, n ≥ 2. Shortly thereafter Prasad's result for ergodicity was extended by Alpern [12, 14] to all properties generic for automorphisms of an infinite Lebesgue space, but still only on the particular manifold Rn. This chapter is devoted to explaining and proving these results for Rn. Unlike the compact case, where the analysis for the cube In is essentially the same as for all compact manifolds, in the noncompact case the study of Rn is directly applicable only to the special class of noncompact manifolds with a single end. However, some of the ideas used here will be of general use for the noncompact setting, so this chapter will give the reader a gentle introduction to the more varied manifolds to come later.

The interest in dynamics on Euclidean space Rn dates at least to the famous Scottish Book [85] of 1935. This was a record kept by Polish mathematicians including Banach, Steinhaus, and Ulam, of problems discussed at the ‘Scottish Cafe’ in Lwów, Poland. Problem 115 of the Scottish Book, posed by Ulam, asks the following question:

In our terminology, this questions asks whether there is a transitive homeomorphism of Rn.

Introduction to Part II

Up to now we have restricted our attention to volume preserving homeomorphisms of the cube, and have proved a number of results for this space M[In, λ]. In this part of the book (Chapters 9 and 10) we show how the results already obtained for M[In, λ] apply more generally to the space M[X, μ] whenever X is any compact connected manifold (we allow situations where our manifold X could possibly have nonempty boundary as for example when X = In) and μ belongs to a certain class of finite measures. In other words, we will show that there was really no loss of generality in restricting our attention to the cube with volume measure, where the intuition was clearer.

We note for later purposes that the situation is very different for noncompact manifolds, in that results obtained for the ‘standard noncompact manifold’ Rn do not go over unchanged to arbitrary noncompact manifolds. That is, for compact manifolds the topological type of the manifold is irrelevant, but for noncompact manifolds the end structure is important. But these are matters to be dealt with in Part III.

General Measures on the Cube

We begin our analysis by retaining for the moment the cube In, n ≥ 2, as our manifold, but now endowing it with a more general Borel probability measure μ.

Introduction

A central idea of real variable theory, ‘Littlewood's Second Principle’, is that every measurable function is nearly continuous. Two forms of this principle are contained in the following well known result, the stronger second part of which is known as ‘Lusin's Theorem’.

Theorem 6.1Let g : R → R be a measurable real valued function with |g(x) – x| < ∈ on the interval [a,b]. Then for any δ > 0 there is a continuous function h : R → R with |h(x) – x| < ∈ on [a,b] satisfying

1. λ {x : | g(x)− h(x)| ≥ δ} < δ, and even

2. λ {x : g(x)≠ h(x)} < δ.

In this chapter we will prove an analogous result which relates measurable and continuous ergodic theory. That is, we show that a volume preserving bimeasurable bijection of the cube In is nearly a volume preserving homeomorphism. The notion of ‘nearly’ is made precise in the following result obtained by Alpern [8].

Theorem 6.2 (Measure Preserving Lusin Theorem)Let g be a bimeasurable volume preserving bijection (i.e., automorphism) of the cube In, n ≥ 2, with ∥g∥ ≡ ess sup |g(x)−x| < ∈. Then given any δ > 0, there is a volume preserving homeomorphism h of In, with ∥h∥ < ∈ and equal to the identity on the boundary of In, satisfying

1. λ {x : |g(x)−h(x)| ≥ δ} < δ

2. λ {x : g(x) ≠ h(x)} < δ.

Introduction

In the previous chapter, we defined the space M[X, μ] of all homeomorphisms of a compact connected manifold X which preserve an OU probability measure μ. In addition, we proved the existence of a ‘Brown map’ ø : In → X, and used it to prove (Theorem 9.7) that typical measure theoretic properties V are also typical in the subspace M[X, ø (∂In), μ] of M[X, μ] consisting of homeomorphisms which pointwise fix the singular set K = ø(∂In). In the next section of this chapter we will show (Theorem 10.3) that this genericity result holds for the full space M[X, μ], although it cannot be established by simple bootstrapping arguments involving the Brown map.

The final section of this chapter considers the existence of fixed points for volume preserving homeomorphisms of the open unit n-cube. Recall that we proved earlier (Theorem 5.5) Montgomery's observation that for n = 2 all such homeomorphisms which are orientation preserving have a fixed point. We will negatively answer the question of Bourgin as to whether Montgomery's result can be extended to higher dimensions or to orientation reversing homeomorphisms. The main tool will be the Homeomorphic Measures Theorem (Theorem 9.1), stated in the previous chapter.

This monograph covers the authors' work over the past twenty five years on generalizing the classical results of John Oxtoby and Stan Ulam on the typical dynamical behavior of manifold homeomorphisms which preserve a fixed measure. In the main text of the book we will take a logical rather than historical perspective, designed to give the reader a concise and unified treatment of results we obtained in a series of articles that were written before the overall structure of the theory was clear. However, since the true significance of this field of study can be understood only from a historical perspective, we devote this preface to a discussion of the problem considered by Oxtoby and Ulam when they were Junior Fellows at Harvard in the 1930s, and of their accomplishment in its solution. We shall use their own words where possible.

The origins of Ergodic Theory lie in the study of physical systems which evolve in time as solutions to certain differential equations. Such systems can be initially described by parameters giving the states of the system as points in Euclidean n-space. Taking conservation laws into account, the phase space may be decomposed into lower dimensional manifolds. Regularities in the differential equations obeyed by the system are reflected in the differentiability or the continuity of the flow that describes the evolution of the system over time. Furthermore, Liouville's Theorem ensures that for Hamiltonian systems this flow has an invariant measure.