Recurrence in Ergodic Theory and Combinatorial Number Theory

HARRY FURSTENBERG
Series: Porter Lectures
Pages: 216
https://www.jstor.org/stable/j.ctt7zv9zv

1. Front Matter
(pp. i-iv)
(pp. v-viii)
3. Foreword from the Porter Lectures Committee
(pp. ix-x)
S. B.

Milton Brockett Porter, after whom the M. B. Porter Lectures are named, was born on November 22, 1869, in Sherman, Texas, and he died, at the age of ninety, on May 27, 1960, in Austin, Texas. In 1897 he received a Ph.D. in mathematics from Harvard University where he enjoyed the attention of the renowned analysts Bôcher, Byerly, and Osgood, and in the year 1901–1902 he was assistant professor at Yale University. In 1902 he was appointed Professor of Pure Mathematics at the University of Texas at Austin, and he remained in this position until becoming emeritus in 1945....

4. PREFACE
(pp. xi-2)
5. INTRODUCTION
(pp. 3-16)

Theoretical dynamics begins with Newton who singlehandedly set it into motion by his development of calculus, his formulation of the laws of motion, and his discovery of the universal law of gravitation. In Newton’s formulation the motion of a dynamical system is governed by a system of differential equations satisfied by the parameters of the system varying in time, and for two centuries following the publication of Newton’sPrincipiathe subject was pursued as a chapter in the theory of differential equations. Most challenging was the problem of applying Newtonian theory to planetary motion, or, more generally, then-body problem...

6. PART I. RECURRENCE IN DYNAMICAL SYSTEMS
• CHAPTER 1 Recurrence and Uniform Recurrence in Compact Spaces
(pp. 19-39)

In this chapter we shall discuss the simplest versions of the notion of recurrence. In the course of our discussion we shall develop some of the basic concepts of topological dynamics. Throughout our discussion adynamical systemwill consist of a compact metric spaceXtogether with a group or semigroupGacting onXby continuous transformations. To avoid problems that are not germane to our point of view, we shall assumeGis discrete and abelian. Most of the time, but not exclusively, we takeGto be the integersZor the natural numbersNwith their...

• CHAPTER 2 Van der Waerden’s Theorem
(pp. 40-56)

In this chapter we shall extend Birkhoff’s recurrence theorem, Theorem 1.1, to the situation where several commuting transformations act on a compact spaceX. We shall show that if${T_1},{T_2},...,{T_l}$are all continuous maps of the compact metric spaceXto itself, there exists some$x \in X$and some sequence${n_k} \to \infty$, with$T_i^{nk}x \to x$simultaneously fori= 1,2,...,l. (Theorem MBR of the Introduction.)

Let us begin by observing that if the condition of commutativity is omitted, the conclusion of this theorem need not hold. One can even arrange that the set of recurrent points for one transformation be disjoint from that...

7. PART II. RECURRENCE IN MEASURE PRESERVING SYSTEMS
• CHAPTER 3 Invariant Measures on Compact Spaces
(pp. 59-78)

Throughout Part II we shall be concerned withmeasure spacesby which we mean a triple (X,B,μ) whereXis a space,Ba σ-algebra of sets inX,μa nonnegative σ-additive measure onXwith$\mu (X) < \infty$. It will sometimes be convenient to normalize the measure so that$\mu (X) = 1$and we shall assume this to be the case from the start. The expression${L^p}(X,B,\mu )$denotes the usual Lebesgue space of measurable functions whosepth power is integrable,$1 \le p \le \infty$, and${L^\infty }(X,B,\mu )$denotes the space of essentially bounded functions. We shall not be careful to distinguish between the space of functions...

• CHAPTER 4 Some Special Ergodic Theorems
(pp. 79-97)

In this chapter we shall present two special cases of the multiple recurrence theorem for commuting measure preserving transformation (Theorem 7.15, quoted in Chapter 3, §7). In these cases we shall be able to compute the average

(4.1)$\mathop {\lim }\limits_{N \to \infty } \frac{1}{{N + 1}}\sum\limits_{n = 0}^N {\mu ({A_0} \cap T_1^{ - n}{A_1} \cap ... \cap T_1^{ - n}{A_l})}$

for any measurable sets${A_0},{A_1},...,{A_l}$in the measure space (X,B,μ). Specializing to the case${A_0} = {A_1} = \cdot \cdot \cdot = {A_l} = A$with$\mu (A) > 0$, we shall find that the limit in (4.1) is positive, and this will establish the assertion of the multiple recurrence theorem in these cases.

In what follows we shall frequently consider the product of measure preserving systems. If$({X_1},{B_1},{\mu _1},{T_1})$and$({X_2},{B_2},{\mu _2},{T_2})$are two m.p.s.,...

• CHAPTER 5 Measure Theoretic Preliminaries
(pp. 98-116)

In this chapter we have collected the technical measure-theoretic material that will be needed in the next two chapters. Most of this material relates to the notion of a factor of a measure space, a notion that will be used repeatedly in the sequel.

We recall the definition of a measure space as a triple (X,B,μ) whereXis an arbitrary space,Bis a σ-algebra of subsets ofX, μis a σ-additive non-negative measure on the sets ofBwith (for convenience)$\mu (X) = 1$. If (Y,D,v) is another measure space, a map$\phi :X \to Y$ismeasure preservingif

(i) it...

• CHAPTER 6 Structure of Measure Preserving Systems
(pp. 117-139)

The objective in the next two chapters is to prove the multiple recurrence theorem for commuting measure preserving transformations of a measure space. In Chapter 4 we established two special cases of this theorem, Theorem 4.12 and Theorem 4.27. If we examine the proofs of these special cases, we see that two distinct phenomena are exploited to obtain recurrence. In the case of a totally weak mixing group of transformations the various terms$T_i^{ - n}A$in the multiple intersection tend to become independent of one another, and$\mu (A \cap T_1^{ - n}A \cap \cdot \cdot \cdot \cap T_l^{ - n}A)$behaves, on the average, like$\mu {(A)^{l + 1}}$. In the proof of Theorem 4.27, the...

• CHAPTER 7 The Multiple Recurrence Theorem
(pp. 140-154)

In this chapter we shall show how the multiple recurrence property can be deduced for an arbitrary system using the structure theorem of the last chapter. We use the transfinite composition series for the system and move up the ladder by arguing that the property in question is preserved both under passage to the limit and under primitive extensions. Most of our work—and this is the heart of the proof—is devoted to the latter, namely, to showing that a primitive extension of a system with the multiple recurrence property also has the multiple recurrence property.

Definition 7.1Let...

8. PART III. DYNAMICS AND LARGE SETS OF INTEGERS
• CHAPTER 8 Proximality in Dynamical Systems and the Theorems of Hindman and Rado
(pp. 157-174)

In this chapter we return to topological dynamics with a discussion of the notion of proximality that we shall tie together with that of recurrence and then apply to combinatorial questions. The main combinatorial consequences are the theorems of Hindman and Rado mentioned in the Introduction. The combinatorial aspects of certain large sets of integers of which Hindman’s theorem is an example will be seen in the next chapter to have implications for topological dynamics and ergodic theory. Our goal has been to focus on the interaction between the two areas, and there will be little by way of new...

• CHAPTER 9 The Fine Structure of Recurrence and Mixing
(pp. 175-194)

In this chapter we continue to study the interplay between recurrence properties of dynamical systems and combinatorial properties of sets of integers. Given a notion of a “large” set of integers, we associate with it a form of recurrence, saying that a point x of a dynamical system (X, T) displays this form of recurrence if for each neighborhoodVof x, the set of return times of x toVis a large set. Thus the notion of a syndetic set leads to that of a uniformly recurrent point. Starting with two other classes of large sets, we shall...

9. BIBLIOGRAPHY
(pp. 195-200)
10. INDEX
(pp. 201-202)
11. Back Matter
(pp. 203-203)