Pointwise Ergodic Theorems for Nonconventional L1 Averages by Patrick LaVictoire A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor F. Michael Christ, Chair Professor Daniel Tataru Assistant Professor Noureddine El Karoui Spring 2010 Pointwise Ergodic Theorems for Nonconventional L1 Averages Copyright 2010 by Patrick LaVictoire 1 Abstract Pointwise Ergodic Theorems for Nonconventional L1 Averages by Patrick LaVictoire Doctor of Philosophy in Mathematics University of California, Berkeley Professor F. Michael Christ, Chair A topic of classical interest in ergodic theory is extending Birkhoff's Pointwise Ergodic Theorem to various classes of nonconventional ergodic averages. Previous methods had only established the pointwise behavior of many such averages when applied to functions in Lp with p > 1, leaving open the important case of L1. In this thesis, I adapt and extend the techniques of Urban and Zienkiewicz [30] and Buczolich and Mauldin [11], whose recent L1 results have renewed interest in the subject, in order to prove several new results on the pointwise convergence of nonconventional averages of L1 functions. The first result (Theorem 1.3) proves that averages along certain Bernoulli random sequences (generated by independent f0; 1g-valued random variables) satisfy a pointwise ergodic the- orem in L1, with probability 1. The second (Theorem 1.4) proves that averages along nd for d ≥ 2, or along the sequence of primes, do not satisfy a pointwise ergodic theorem in L1. (The result for n2 was first shown in [11]; the others are new.) The third (Corollary 1.9) shows that, given a Fourier decay condition on a sequence of probability measures on Z, some subsequence of the corresponding nonconventional averages will satisfy a pointwise ergodic theorem in L1. i Contents 1 Introduction 1 1.1 Context: Ergodic Theorems . 1 1.2 Random Subsequences of Density 0 . 4 1.3 Universally L1-Bad Arithmetic Sequences . 5 1.4 Sequences of Convolution Operators . 6 1.5 The Banach Principle and the Conze Transference Principle . 7 1.6 Two Recurring Tools . 10 1.7 L2 Theory . 11 1.8 L1 Theory: Heuristics . 13 2 Sequences of Convolution Operators 16 2.1 Approximate Identities . 16 2.2 Weighted Averages . 18 2.3 Non-Uniform Fourier Decay . 22 3 Random Subsequences of Density 0 23 3.1 Reductions . 23 3.2 Calder´on-Zygmund Argument . 24 3.3 Probabilistic Lemma . 28 4 Universally L1-Bad Arithmetic Sequences 32 4.1 Introduction . 32 4.2 Outline of the Argument for the Squares . 33 4.3 Main Theorem . 36 4.4 The Inductive Step . 39 4.5 Periodic Rearrangements . 42 L+1 4.6 Defining fK ................................... 45 γ 4.7 Restricting a Family to (−Λq) ......................... 46 4.8 Completion of the Inductive Step . 48 4.9 Notes on the Proof . 52 ii Bibliography 54 iii Acknowledgments The author thanks his thesis advisor, M. Christ, for years of invaluable ideas, advice, heuris- tics, and direction; J. Rosenblatt and M. Wierdl for many conversations, suggestions and encouragement, particularly on the results in Sections 2 and 4 (respectively); many other mathematicians whose correspondence and support have been greatly helpful at various points, including C. Demeter, C. Thiele, M. Lacey, B. Stovall, and K. Datchev; the authors F. Dostoyevsky, F. Nietzsche and D. Hofstadter, among others; his extraordinary friends and family; and you, for reading this. 1 Chapter 1 Introduction 1.1 Context: Ergodic Theorems The starting point for this thesis is the Birkhoff pointwise ergodic theorem, as we will be discussing various extensions of this to other weighted ergodic averages. Definition A dynamical system (X; F; m; T ) consists of a topological space X with σ- algebra F, a probability measure m on (X; F), and a measure-preserving transformation T : X ! X; that is, m(T −1A) = m(A) 8A 2 F. We will usually suppress F in all that follows. Definition A dynamical system (X; F; m; T ) is ergodic if it is non-atomic (i.e. 8x 2 X; m(fxg) = 0) and if every T -invariant set A = T −1A 2 F has measure 0 or 1. Example Let X = T = R=Z, with Lebesgue measure and the measure-preserving transfor- mation Tαx = x + α (mod 1). If α2 = Q, the Fourier transform reveals that any Tα-invariant 2 function f = f ◦ Tα 2 L (T) must be constant a.e, and therefore the dynamical system is p ergodic. If α = q , however, the set A = fx 2 T :(qx mod 1) 2 [0; 1=2)g is a nontrivial Tα-invariant set. Example Let X = ZN , with the counting measure (rescaled by 1=N to be a probability measure) and the measure-preserving transformation T k = k + 1 (mod N). Note that if N is prime, this dynamical system has no nontrivial T -invariant sets. The notion of an ergodic dynamical system is defined in order to exclude this example, because as we shall see, there is behavior shared by all ergodic dynamical systems but not by such systems as this. Example Let Let X = T = R=Z, with Lebesgue measure and the \doubling map" T x = 2x (mod 1). While this map is two-to-one (however, note Remark 5 below), it indeed has the property that m(T −1A) = m(A) 8A 2 F, and is a well-studied ergodic transformation (again, consider the Fourier transform of the characteristic function of an invariant set). CHAPTER 1. INTRODUCTION 2 There are two very different classical versions of the ergodic theorem, proved by Von Neu- mann and Birkhoff in the 1930s: Theorem 1.1 (Von Neumann Mean Ergodic Theorem). Let (X; F; m; T ) be a dynamical N 1 X system, and f 2 L2(X; m). Then the averages A f(x) := f(T kx) converge in the L2 N N k=1 norm. If (X; F; m; T ) is ergodic, then the limit of these averages is the constant function Z f dm. X Theorem 1.2 (Birkhoff Pointwise Ergodic Theorem). Let (X; F; m; T ) be a dynamical sys- N 1 X tem, and f 2 L1(X; m). Then the averages A f(x) := f(T kx) converge for almost N N k=1 every x 2 X (by the measure m). If (X; F; m; T ) is ergodic, then the a.e. limit of these Z averages is the constant f dm. X We will be considering extensions of ergodic theorems to sequences of \nonconventional" 1 weighted ergodic averages. For any sequence of complex numbers faN;jgN2 ;j2 such that P N Z jaN;jj < 1 for all N, we may define for any dynamical system (X; F; m; T ) and any j2Z f 2 L1(X; m) the weighted ergodic averages X j AN f(x) := aN;jf(T x); (1.1) j2Z and we are interested in whether this sequence of weighted averages converges (in the mean or a.e.) for all dynamical systems and all functions in a given Lp class. Remarks: Z 1. The mean or pointwise limit for a sequence of weighted averages need not equal f dm X if it exists. We will only be concerned with whether such a limit exists in the first place. 2. A pointwise ergodic theorem for a sequence of weighted ergodic averages easily implies the corresponding mean ergodic theorem, via the Dominated Convergence Theorem; a mean ergodic theorem does not imply a pointwise ergodic theorem. 3. A mean ergodic theorem is equivalent to the same statement with L2 replaced by Lp, for any 1 ≤ p < 1. 1For such general arrays, it is an abuse of notation to call the objects (1.1) \averages"; but in all that follows (save for a remark at the end of Chapter 2), we will be concerned only with averages corresponding to probability measures on Z. CHAPTER 1. INTRODUCTION 3 4. A pointwise ergodic theorem for Lp functions implies one for Lq functions if and only if p ≤ q. For example, Bellow [4] constructed sequences for any p > 1 so that the corresponding weighted averages have pointwise ergodic theorems for Lp but not for any Lq with q < p. 5. It may seem illegitimate to allow negative powers of T in weighted ergodic averages. However, for our purposes, we do not lose any generality by restricting ourselves to invertible transformations T , as the sequence ff ◦ T n : n ≥ 0g is always equal in joint distribution to the corresponding sequence for some invertible transformation. See for example [22], Theorem 4.8. A bit of notation will be useful for several of the results in this thesis. One important type of nonconventional ergodic average concerns the average along a sequence of integers fnkg; for any (X; F; m; T ) and any f 2 L1(X; m), we can define the subsequence averages N fn g 1 X A k f(x) := f(T nk x): (1.2) N N k=1 p Definition We say that fnkg is universally L -good if for every dynamical system (X; F; m; T ) p fnkg and every f 2 L (X; m), lim AN f(x) exists for almost every x 2 X. We say that fnkg N!1 is universally Lp-bad if for every ergodic dynamical system (X; F; m; T ); there exists an p fnkg 1 f 2 L (X; m) such that the sequence fAN f(x)gN=1 diverges on a set of positive measure in X. p k It is easy to construct sequences which are universally L -bad; for instance, nk = 2 (or indeed, any lacunary sequence of natural numbers) is universally L1-bad (see [23] or [1]).