Recurrence in Ergodic Theory and Combinatorial Number Theory

by Harry Furstenberg

Overview

Topological dynamics and ergodic theory usually have been treated independently. H. Furstenberg, instead, develops the common ground between them by applying the modern theory of dynamical systems to combinatories and number theory.

Originally published in 1981.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Product Details

ISBN-13:	9780691615363
Publisher:	Princeton University Press
Publication date:	07/14/2014
Series:	Porter Lectures , #119
Pages:	216
Product dimensions:	6.90(w) x 9.90(h) x 0.60(d)

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Recurrence in Ergodic Theory and Combinatorial Number Theory

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By Harry Furstenberg

PRINCETON UNIVERSITY PRESS

Copyright © 1981 Princeton University Press
All rights reserved.
ISBN: 978-0-691-08269-1

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CHAPTER 1

Recurrence and Uniform Recurrence in Compact Spaces

1. Dynamical Systems and Recurrent Points

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 a dynamical system will consist of a compact metric space X together with a group or semigroup G acting on X by continuous transformations. To avoid problems that are not germane to our point of view, we shall assume G is discrete and abelian. Most of the time, but not exclusively, we take G to be the integers Z or the natural numbers N with their additive structure. In general we denote a dynamical system by (X,G). When G is either Z or N, so that it is generated by the element 1, we denote by T the transformation on X representing the action of the element 1. We then denote the dynamical system by (X,T) and call it a cyclic system. When no confusion exists, we identify G with the set of transformations of X corresponding to the given action. Thus while we should denote the transformation corresponding to g [member of] G by Tg: X [right arrow] X, we usually write gx instead of Tgx, except in the cyclic case where we denote T1x by Tx.

The point of departure for our discussion is the following definition.

Definition 1.1. Let T be a continuous map of a topological space X into itself. A point x [member of] X is called a recurrent point for T (or for the dynamical system (X,T)) if for any neighborhood V [contains as member] x, there exists n ≥ 1 with Tn x [member of] V.

When X is a metric space, we can let V range through a sequence of neighborhoods of diameter [right arrow] 0, and we conclude that for some sequence nk, Tnk x [right arrow] x. This could be taken, in the metric case, to be the definition of recurrence.

Using Zorn's Lemma, we can prove that if X is a compact space, then for any continuous map T : X [right arrow] X, there always exist recurrent points. (In Chapter 3 we shall give another proof that doesn't use Zorn's lemma and also gives somewhat more.)

Suppose X is compact and consider the family F of closed subsets φ ≠ Y [subset] X satisfying TY [subset] Y and ordered by inclusion. We claim F has a minimal element. Namely if we have a totally ordered chain of such subsets, their intersection is nonempty and again a set in F. Hence Zorn's lemma applies to produce a minimal element of F. Say Y0 is minimal. We claim each point of Y0 is recurrent. For let x [member of] Y0 and consider Y = {Tnx, n ≥ 1}. We call this the forward orbit closure of x. Now Y [subset] Y0 because Y0 is closed and invariant under T. But so is Y and so, by minimality, Y= Y0. That means that each neighborhood of x contains some Tnx for n ≥ 1. We have proved:

Theorem 1.1. If T is a continuous map of a compact space X to itself, the set of recurrent points for T in X is nonempt y.

The simplest example of the phenomenon of recurrence and a direct generalization of periodicity is that of a Kronecker system.

Definition 1.2. Let K be a compact group, let a [member of] K, and Let T: K [right arrow] K be defined by Tx = ax, left multiplication with the element a. We then call the dynamical system (K, T) a Kronecker system.

The reason for the terminology is that one can reformulate a well-known theorem of Kronecker to assert that when K is a torus, the identity is a recurrent point of the corresponding Kronecker system. More generally we have

Theorem 1.2. Every point in the space of a Kronecker system is recurrent.

Proof: Some point x0 is recurrent. Now let x be any point and write x = x0u. If V is a neighborhood of x, then Vu-1 is a neighborhood of x0 and anx0 [member of] Vu-1 implies an(x0u) [member of] V.

2. Automorphisms and Homomorphisms of Dynamical Systems, Factors, and Extensions

Let (X,G) and (Y,G) be two dynamical systems with the same (semi-) group G of operators. A homomorphism from (X,G) to (Y,G) is given by a continuous map φ: X [right arrow] Y satisfying

(1.1) φ (gx) = gφ(x)

for x [member of] X, g [member of] G. Note that the products gx and g φ(x) refer to the action of G on two different spaces.

DEFINITION 1.3. A dynamical system (Y,G) is a factor of the dynamical system (X,G) if there is a homomorphism of the latter to the former given by a map φ of X onto Y. In this case we also say that (X,G) is an extension of (Y,G).

Assume φ: X [right arrow] Y is onto. On account of the compactness of the spaces we can identify Y with the set of fibers {φ-1(y): t [member of] Y} of the map φ, appropriately topologized. Equation (1.1) implies that the action of G is compatible with this identification. Thus all the information regarding a factor of a system is implicit in the system. In particular, phenomena taking place on (X,G) generally carry over to factors of (X,G). Here is a simple example.

Proposition 1.3. If φ determines a homomorphism of a cyclic system (X,T) to (Y,T) and x [member of] X is recurrent for (X,T), then φ(x) is recurrent for (Y,T).

We shall now give a class of examples where we can invert the order of the implication and assert that the preimage of a recurrent point is recurrent. This takes place in the case of a group extension, a notion that will play an important role in the sequel.

Definition 1.4. Let (Y,T) be a dynamical system, K a compact group, and ψ : Y [right arrow] K a continuous mapping. Form X = Y × K and define T : X [right arrow] X by T(y,k) = (Ty, ψ(y)k). The resulting system (X,T) is called a group extension of (Y,T), or, sometimes, a skew product of (Y,T) with K.

Of course, the map (y,k) [right arrow] y defines a homomorphism of (X, T) to (Y,T) so that (X,T) is an extension of (Y,T).

If (X,T) = (Y × K, T) is a group extension, then the elements of K act on X by right translation:

(1.2) Rk,(y,k) = (y,kk'),

and the Rk' commute with T. As a result the Rk' constitute automorphisms of the system (X, T). This fact will be useful in proving the next theorem.

Theorem 1.4. If y0 [member of] Y is recurrent for (Y,T), and (X, T) is a group extension of (Y,T), X = Y × K, then each of the points (y0, k0), k0 [member of] K, is a recurrent point of (X,T).

Proof: Let e denote the identity of K. We shall show that (y0, e) is a recurrent point of (X,T), and then it follows that each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is recurrent, since the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are automorphisms. For any x [member of] X, let us denote by Q(x) the forward orbit closure: Q(x) = {Tnx : n ≥ 1}. Now x is recurrent [??] x [member of] Q(x). Since y0 is recurrent for (Y,T), some (y0, k1) [member of] Q(y0, e). Apply the automorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to this inclusion and recall that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and we find (y0,k21) [member of] Q(y0, k1) Now the relationship x' [member of] Q(x) is transitive, and so we find (y0, k21) [member of] Q(y0,e). Applying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] again and repeating, we find successively [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] But now by Theorem 1.2 and the fact that Q(y0,e) is closed, we conclude (y0,e) [member of] Q(y0,e).

Using Theorem 1.4, we can inductively obtain examples of non-Kronecker dynamical systems on a torus for which every point is recurrent. For example, consider the 2-torus

T2 = {(θ, φ): θ, φ [member of] R/Z}

where the components are taken as reals modulo 1. Let T: T2 [right arrow] T2 be given by T(θ, φ) = (θ + α, φ + 2θ + α). Then (T2, T) is a group extension of the Kronecker system on the circle: T θ = θ + α ψ: T [right arrow] T given by ψ (θ) = 2θ + α. By Theorems 1.2 and 1.4, every point of T2 is recurrent. Let us compute the orbit of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], as one verifies by induction. The fact that this comes abritrarily close to (0,0) yields the following proposition.

Proposition 1.5. For any real number a and for any ε > 0, we can solve the diophantine inequality |αn2 - m| < ε.

This also indicates that for any real number α there are "good" rational approximations using only rationals with a square in the denominator:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proposition 1.5 was first proved by Hardy and Littlewood [1]. Their proof was superseded by Weyl's more powerful method of trigonometric sums (Weyl [1]). We shall retrieve most of Weyl's results in the course of our discussion.

We can extend Proposition 1.5 to polynomials of higher degree. Let p(x) be a polynomial of degree d with real coefficients. Write pd(x) = p(x) and form successively

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

etc. Each pi(x) is of degree i; let p0(x) be the constant [varies]. Now define a transformation of the d-dimensional torus Td] [right arrow] Td] by

(1.3) T(θ1, θ2, θ1, ..., θd = (θ1 + [varies], θ2 + θ1, θ3 + θ2, ..., θd + θd-1).

Again, this will define a group extension of a dynamical system on the (d - 1)-torus, which in turn is a group extension of a dynamical system on the (d-2)-torus, etc., down to the 1-torus. We conclude that each point is recurrent. Now compute the orbit of the point (p1(0), p2(0), ... pdd(0)). Since pi(n) + pi-1(n) = pi(n + 1) we find that

[T(p1(n),p2(n), ..., pd(n)) = (p1(n + 1), p2(n + 1), ..., pd(n + 1))

and

Tn (p1(0), p2(0), ..., pd(0) = (p1(n),p2(n), ..., pd(n).

We conclude that pd(n) = p(n) comes arbitrarily close to p(0) modulo 1. Thus the foregoing proposition has the following generalization.

Theorem 1.6. If p(x) is any real polynomial with p(0) = 0, then for any ε > 0 we can solve the diophantine inequality

(1.4) |p(n) - ml < ε, n> 0.

In the foregoing we have applied Theorem 1.4 to the special case where K is the circle group and φ is a rather special function. These are the only cases for which we can formulate explicit results, as in Theorem 1.6, but in principle there is a much wider range of applications.

It is sometimes useful to broaden the notion of a group extension to that of an isometric extension.

Definition 1.5. Let K be a compact group of isometries of a compact metric space M , where K is topologised so that the map K × M [right arrow] M is continuous. Let (Y,T) be a dynamical system and ψ : Y [right arrow] K continuous. Form X = Y × M and define T(y,u) = (Ty,ψ(y)u), where ψ(y)u denotes the image of u under the isometry of M determined by ψ(y) [member of] K. The system (X,T) is an isometric extension of (Y,T).3

Thus M might be the sphere Sn-1 or the ball Bn [subset] Rn and K the orthogonal group O(n).

Proposition 1.7. If (X,T) is an isometric extension of (Y,T), then X = [universal] Xα, where Xα is a closed T-invariant subset of X and the system (Xα, T) is a factor of a group extension of (Y,T).

Proof: Let X = Y × M and let ψ: Y [right arrow] K as in Definition 1.5. Form the product [bar.X] = Y × K and define T: [bar.X] [right arrow] [bar.X] using the same function ψ : T(y,k) = (Ty, ψ(y)k ). Now let (y0, u0) be any point of X and define ψ : [bar.X] [right arrow] X by ψ(y,k) = (y,ku0). It is easy to see that φ defines a homomorphism of ([??],T) onto a subsystem of ([bar.X],T) and that (y0, u0) = φ(y,e) is in the image of this homomorphism. This proves the proposition.

As an immediate corollary we obtain

Theorem 1.8. If (X,T) is an isometric extension of (Y,T), then the preimage of any recurrent point of (Y,T) is recurrent for (X,T).

3. Recurrent Points for Bebutov Systems

Let G be a countable (semi-) group and Λ a compact metric space. Form Ω = ΛG, the compact metrizable space of all functions from G to Λ. We define an action of G on Ω (the regular action) by letting

g'ω(g) = ω(gg'), ω [member of] Ω, g,g' [member of] G.

The reader should convince himself that this is an action, i.e., that g'(g"ω) = (g'g")ω.

Definition 1.6. A Bebutov system is a subsystem of (Q,G), i.e., it is a system (X,G) where X [subset] Ω is a closed subset invariant under the regular action of G. A symbolic flow is a Bebutov system for which Λ is finite and G is eitherNorZ.

In the case of a symbolic flow the points of Ω can be thought of as sequences of symbols. Sometimes we call Λ the alphabet.

It is sometimes useful to fix a metric on Ω. If d (·,·) denotes the metric on Λ and G = {g1, g2, ...} is an enumeration of the elements of G, one can define a metric on Ω by

(1.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Equation (1.5) implies that two points ω,ω [member of] Ω will be close if their values are close on "large" sets in G. In particular if Λ is discrete and G = N, then the sequences are close if they agree for a large block of numbers 1,2, ..., N. If they disagree at 1 but agree at all other entries, they will still be far apart.

If ω0 [member of] Ω, we can form the smallest G-invariant closed subset containing ω0; this is the orbit closure of ω0 in Ω and we denote it Xω0. The system (Xω0, G) will be referred to as the dynamical system or the Bebutov system generated by ω0.

We state without proof the following theorem that ties together the theory of almost periodic functions with the theory of Kronecker systems.

(Continues...)

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Table of Contents

• FrontMatter, pg. i
• CONTENTS, pg. v
• Foreword from the Porter Lectures Committee, pg. ix
• Preface, pg. xi
• Introduction, pg. 1
• Chapter 1. Recurrence and Uniform Recurrence in Compact Spaces, pg. 19
• Chapter 2. Van der Waerden's Theorem, pg. 40
• Chapter 3. Invariant Measures on Compact Spaces, pg. 59
• Chapter 4. Some Special Ergodic Theorems, pg. 79
• Chapter 5. Measure Theoretic Preliminaries, pg. 98
• Chapter 6. Structure of Measure Preserving Systems, pg. 117
• Chapter 7. The Multiple Recurrence Theorem, pg. 140
• Chapter 8. Proximality in Dynamical Systems and the Theorems of Hindman and Rado, pg. 157
• Chapter 9. The Fine Structure of Recurrence and Mixing, pg. 175
• Bibliography, pg. 195
• Index, pg. 201