We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
To send content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about sending content to .
To send content items to your Kindle, first ensure no-reply@cambridge.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about sending to your Kindle.
Note you can select to send to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
The symmetric rendezvous problem on a network Q asks how two players, forced to use the same mixed strategy, can minimize their expected meeting time, starting from a known initial distribution on the nodes of Q. This minimum is called the (symmetric) ‘rendezvous value’ of Q. Traditionally, the players are assumed to receive no information while playing the game. We consider the effect on rendezvous times of giving the players some information about past actions and chance moves, enabling each of them to apply Bayesian updates to improve his knowledge of the other's whereabouts. This technique can be used to give lower bounds on the rendezvous times of the original game (without any revealed information). We consider the case in which they are placed a known distance apart on the line graph Q (known as ‘symmetric rendezvous on the line’). Our approach is to concentrate on a general analysis of the effect of revelations, rather than compute the best bounds possible with our technique.
First we describe the work of the first author leading to the conclusion that any property generic (in the weak topology) for measure-preserving bijections of a Lebesgue probability space is also generic (in the compact-open topology) for homeomorphisms of a compact manifold preserving a fixed measure. Then we describe the work of both authors in extending this result to non-compact manifolds, with modifications based on the ends of the manifold. These results can be thought of as generalizations of the original work which established genericity for the specific property of ergodicity (Oxtoby and Ulam, 1941) and subsequent work for other properties such as weak mixing (Katok and Stepin, 1970). The techniques used to obtain the titled theorem are also applied to related areas, such as fixed point theorems and chaos theory, and some new results are obtained.
In this chapter we show that any volume preserving homeomorphism of the cube can be uniformly approximated by volume preserving automorphisms (not generally continuous) with certain specified measure theoretic properties. As shown in the previous section (when the property was ergodicity), this approximation can then be combined with the Lusin Theory to produce homeomorphisms possessing that property, and a version of Theorem C (of Chapter 1) for generic properties of volume preserving homeomorphisms of the cube.
Suppose we want to find homeomorphisms of In which have some particular measure theoretic property, such as ergodicity or weak mixing. Such a property can be designated by specifying a subset V of the space G[X, μ] of all automorphisms of a Lebesgue space (X, μ), which we will take for convenience as all volume preserving bijections of (In, λ). We will only consider properties V which don't depend on the names of the points, i.e., which are conjugate invariant in G[In, λ]. (This assumption means that g ∈ V implies f−1gf ∈ V for all f ∈ G[In, λ].) In this context the statement at the beginning of this paragraph is equivalent to showing V ∩ M[In, λ] is nonempty. Many important measure theoretic properties in G[In, λ] are determined by a countable number of conditions, which each define an open set in the weak topology on G[In, λ], that is, they are Gδ subsets of G[In, λ].
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.
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.
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.
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.
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.
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.
Two of the principal analytic structures that may be put on a set X are measure and topology. We are interested in transformations of X which preserve both of these structures: measure preserving homeomorphisms. In the first half of the book, Parts I and II, the topological space X will be a compact manifold, possibly with boundary. (In fact Part I specializes to the case where X is simply the unit cube In in some dimension n ≥ 2.) The measure, denoted µ, will be a nonatomic Borel probability measure which assigns the manifold boundary measure zero and is positive on all nonempty open sets (a property we call locally positive). (In Part I, µ is simply the volume measure on the cube.) The first two parts of the book are concerned with determining typical properties of µ-preserving homeomorphisms of the (arbitrary) compact manifold X. We denote the set of all such homeomorphisms by M[X, µ], which we endow with the uniform topology, with respect to which it is complete. We call a property typical, or generic, if it is possessed by a dense Gδ (or larger) subset of transformations. The purpose of this introductory chapter is to give a nontechnical presentation of the main results, and the definitions they involve, for measure preserving homeomorphisms of compact manifolds. Both the definitions and theorems mentioned in this chapter will be presented more rigorously in later chapters.
This 2000 book provides a self-contained introduction to typical properties of homeomorphisms. Examples of properties of homeomorphisms considered include transitivity, chaos and ergodicity. A key idea here is the interrelation between typical properties of volume preserving homeomorphisms and typical properties of volume preserving bijections of the underlying measure space. The authors make the first part of this book very concrete by considering volume preserving homeomorphisms of the unit n-dimensional cube, and they go on to prove fixed point theorems (Conley–Zehnder– Franks). This is done in a number of short self-contained chapters which would be suitable for an undergraduate analysis seminar or a graduate lecture course. Much of this work describes the work of the two authors, over the last twenty years, in extending to different settings and properties, the celebrated result of Oxtoby and Ulam that for volume homeomorphisms of the unit cube, ergodicity is a typical property.
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:
Does there exist a homeomorphism h of the Euclidean space Rn with the following property? There exists a point p for which the sequence of points hn(p) is everywhere dense in the whole space.
In our terminology, this questions asks whether there is a transitive homeomorphism of Rn.