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Ergodic Theorems for Polynomials in Nilpotent Groups

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Ergodic Theorems for Polynomials in Nilpotent Groups Ergodic theorems for polynomials in nilpotent groups ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op donderdag 12 september 2013, te 14:00 uur door arXiv:1309.0345v1 [math.DS] 2 Sep 2013 Pavel Zorin-Kranich geboren te St. Petersburg, Rusland Promotiecommissie Promotor: prof. dr. A.J. Homburg Copromotor: dr. T. Eisner Overige leden: prof. dr. E. Lesigne prof. dr. R. Nagel dr. H. Peters prof. dr. C. Thiele prof. dr. J.J.O.O. Wiegerinck Faculteit der Natuurwetenschappen, Wiskunde en Informatica Contents Contents iii Introduction vii Acknowledgments................................... ix Summary ........................................ x Samenvatting...................................... x 1 Nilpotent groups 1 1.1 Generalfacts................................... 1 1.1.1 Commutators and filtrations . 1 1.1.2 Commensurable subgroups . 4 1.1.3 Hirschlength .............................. 6 1.2 Polynomialmappings.............................. 8 1.3 IP-polynomials .................................. 11 1.3.1 IP-polynomials in several variables . 12 1.3.2 Polynomial-valued polynomials . 13 1.3.3 Monomialmappings.......................... 14 2 Mean convergence 17 2.1 A close look at the von Neumann mean ergodic theorem . 18 2.2 Complexity .................................... 21 2.3 Thestructuretheorem ............................. 24 2.4 Reducible and structured functions . 26 2.5 Metastability of averages for finite complexity systems ......... 28 2.6 Right polynomials and commuting group actions . ..... 35 iii iv CONTENTS 3 Recurrence 39 3.1 Topological multiple recurrence . 40 3.1.1 PETinduction.............................. 41 3.1.2 Nilpotent Hales-Jewett theorem . 42 3.1.3 Multiparameter nilpotent Hales-Jewett theorem . ..... 44 3.2 FVIPgroups ................................... 46 3.2.1 Partition theorems for IP-rings . 47 3.2.2 IP-limits ................................. 49 3.2.3 Generalized polynomials and FVIP groups . 53 3.3 Measurable multiple recurrence . 58 3.3.1 Compactextensions .......................... 58 3.3.2 Mixing and primitive extensions . 60 3.3.3 Existence of primitive extensions . 61 3.3.4 Almostperiodicfunctions . 62 3.3.5 Multiplemixing ............................ 63 3.3.6 Multiparameter multiple mixing . 65 3.3.7 Lifting multiple recurrence to a primitive extension ...... 67 3.3.8 Good groups of polynomial expressions . 71 4 Higher order Fourier analysis 73 4.1 Nilmanifolds and nilsequences . 73 4.1.1 Rationality and Mal’cev bases . 74 4.1.2 Commensurable lattices . 75 4.1.3 Cubeconstruction ........................... 78 4.1.4 Verticalcharacters ........................... 79 4.2 Leibman’s orbit closure theorem . 83 4.2.1 Nilsequences .............................. 83 4.2.2 Reduction of polynomials to connected Lie groups . 86 4.2.3 Equidistribution criterion . 87 4.2.4 Leibman’s orbit closure theorem . 92 4.2.5 Nilsystems................................ 94 4.3 Backgroundfromergodictheory. 95 4.3.1 Følnersequences............................ 95 4.3.2 Lindenstrauss covering lemma . 96 4.3.3 Fullygenericpoints .......................... 98 4.3.4 Ergodicdecomposition .. .. .. .. .. .. .. .. .. .. .. .. 99 4.3.5 Host-Kracubespaces . .. .. .. .. .. .. .. .. .. .. .. .. 100 4.4 Wiener-Wintner theorem for nilsequences . 103 4.4.1 The uniformity seminorm estimate . 104 CONTENTS v 4.4.2 Acounterexample ........................... 108 4.4.3 Wiener-Wintner theorem for generalized nilsequences . 109 4.4.4 L2 convergence of weighted multiple averages . 111 5 Return times theorems 115 5.1 Return times theorem for amenable groups . 116 5.1.1 Theorthogonalitycondition . 117 5.1.2 Self-orthogonality implies orthogonality . 118 5.1.3 Return times theorem for amenable groups . 122 5.2 Multiple term return times theorem . 123 5.2.1 Conventions about cube measures . 123 5.2.2 Return times theorem on cube spaces . 125 5.2.3 A measure-theoretic lemma . 127 5.2.4 Universal disintegration of product measures . 130 5.2.5 The sufficient special case of convergence to zero . 133 5.3 Wiener-Wintner return times theorem for nilsequences . ........135 Bibliography 137 Index 145 Introduction Furstenberg’s groundbreaking ergodic theoretic proof [Fur77] of Szemerédi’s the- orem on arithmetic progressions in dense subsets of integers suggested at least two possible directions for generalization. One is connected with earlier work of Furstenberg and consists in investigating actions of groups other than Z. The other looks at polynomial, rather than linear, sequences. Indeed, in the same article Furstenberg [Fur77, Theorem 1.2] gave a short qualitative proof of Sárközy’s theo- rem [Sár78] on squares in difference sets. Furstenberg’s proof of the multiple recurrence theorem involves three main steps: a structure theorem for measure-preserving systems that exhibits dichotomy between (relative) almost periodicity and (relative) weak mixing, a coloring ar- gument that deals with the almost periodic part of the structure, and a multiple weak mixing argument dealing with the weakly mixing part. It was the structure theorem whose generalization to Zd -actions required most of the additional work in the multiple recurrence theorem for commuting transformations due to Fursten- berg and Katznelson [FK78]. The coloring argument carried through using Gallai’s multidimensional version of van der Waerden’s theorem. Also the multiple weak mixing argument worked similarly to the case of Z-actions, namely by induction on the number of terms in the multiple ergodic average. However, in the polynomial case, induction on the number of terms does not seem to work. This difficulty has been resolved by Bergelson [Ber87] who has found an appropriate induction scheme, called PET induction. Later, jointly with Leibman [BL96], he completed his program by proving a polynomial multiple re- currence theorem by Furstenberg’s method, using a polynomial van der Waerden theorem as the main new ingredient. This is when polynomials in nilpotent groups appeared on the stage. Both the coloring and the multiple weak mixing steps in Furstenberg’s framework involve polynomial maps and PET induction when carried out for nilpotent groups, even if one is ultimately interested in linear sequences, see [Lei98]. A similar phenomenon vii viii INTRODUCTION occurs in Walsh’s recent proof of norm convergence of nilpotent multiple ergodic averages [Wal12] (which we extended to arbitrary amenable groups in [ZK11]). Thus polynomials in nilpotent groups, with Leibman’s axiomatization [Lei02], seem indispensable for understanding multiple recurrence for nilpotent group actions. While the work mentioned above concerns Cesàro averages of multicorrelation sequences, there are by now at least two alternative approaches to the study of asymptotic behavior of dynamical systems: using IP-limits or using limits along idempotent ultrafilters. It is the former approach on which we concentrate. The proof of the IP multiple recurrence theorem due to Furstenberg and Katznelson [FK85] parallels Furstenberg’s earlier averaging arguments, although rigidity re- places almost periodicity, mild mixing weak mixing, and the Hales-Jewett theorem the van der Waerden theorem. A direct continuation of their work in the polyno- mial direction has been carried out by Bergelson and McCutcheon [BM00], mixing the techniques outlined above and a polynomial extension of the Hales-Jewett the- orem proved earlier by Bergelson and Leibman [BL99]. One of our main results is a nilpotent extension of the IP multiple recurrence theorem [ZK12]. We take a somewhat novel approach to the structure theorem, obtaining dichotomy between compactness and mixing not on the level of the act- ing group, but on the level of the group of polynomials with values in the acting group. We have also found it necessary to prove a new coloring result, sharpening the nilpotent Hales-Jewett theorem due to Bergelson and Leibman [BL03]. On the other hand, the mixing part is handled in essentially the usual way using PET in- duction. As remarked earlier, this method compels us to deal with polynomial map- pings. Our arguments heavily rely on an efficient axiomatization of IP-polynomials along the lines of Leibman’s work, but incorporating some more recent ideas. Another reason to study polynomial, rather than linear, sequences in nilpotent groups comes from quantitative equidistribution theory on nilmanifolds (that is, compact homogeneous spaces of nilpotent Lie groups). Nilmanifolds play an im- portant role in the theory of multiple ergodic averages [HK05], where one is inter- ested in linear orbits of the form (gn x), where x is a point in the nilmanifold and g an element of the structure group. An obstacle for establishing results that are uniform in all such orbits is the fact that g is drawn from the possibly non-compact structure group. This can be circumvented by considering polynomial orbits, since every linear orbit on a nilmanifold can be represented as a polynomial orbit with coefficients that come from fixed compact subsets of the structure group. This is an important ingredient in the proof of the quantitative equidistribution theorem of Green and Tao [GT12]. We review this circle of ideas in order to motivate our proof of the uniform extension of the Wiener-Wintner theorem for nilsequences. This is a result about ix
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