NIKOS FRANTZIKINAKIS. N n N where (Φ N) N N is any Følner sequence

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1 SOME OPE PROBLEMS O MULTIPLE ERGODIC AVERAGES IKOS FRATZIKIAKIS. Probems reated to poynomia sequences In this section we give a ist of probems reated to the study of mutipe ergodic averages invoving iterates given by poynomia sequences, and reated appications to mutipe recurrence... Powers of a singe transformation. Let P := {p,..., p } be a famiy of integer poynomias that are essentiay distinct, meaning, a poynomias and their differences are nonconstant. First, we consider averages of the form () T p(n) f... T p(n) f, n= where (X, X, µ, T ) is an invertibe measure preserving system and f,..., f L (µ). We remark that a the mean convergence resuts stated in this section work equay we for averages of the form Φ n Φ in pace of the averages n where (Φ ) is any Føner sequence of subsets of. Before discussing some probems reated to the characteristic factors of the averages (), we state a resut of B. Host and B. Kra [32] and A. Leibman [35] that gives usefu information about their structure. Theorem. There exists d = d(p) such that the factor Z d,t is characteristic for mean convergence of the averages (). We emphasize that the vaue of d(p) in the previous statement does not depend on the system or the functions invoved. Given a famiy of poynomias P, we denote by d min (P) the minima vaue of d(p) that works in the previous theorem. This vaue is in genera hard to pin down and depends on the agebraic reations that the poynomias satisfy. For instance, we know that d min ({n, 2n,..., n}) = ([3, 45]), and d min (P) = when P consists of at east two rationay independent poynomias ([22, 24]). But it is not ony inear reations between the poynomias that matter, for instance, we know that d min ({n, 2n, n 2 }) = 2 whie d min ({n, 2n, n 3 }) = ([2, 36]). More exampes of famiies P where d min (P) has been computed can be found in [2, 36]. Furthermore, in [36] a (rather compicated) agorithm is given for computing this vaue. Despite such progress, the foowing is sti open (the probem is impicit in [] and was stated expicity in [36]): Probem. If P 2, show that d min (P) P. Date: March 20.

2 2 IKOS FRATZIKIAKIS The estimate is known when P = 2, 3 ([2]) and it is open for P = 4. The probem is even open when one is restricted to the cass of Wey systems, meaning, systems of the form (T d, B T d, m T d, T ) where T : T d T d is a unipotent affine transformation. We denote with d W min (P) the minimum vaue of d(p) such that the factor Z d(p),t is characteristic for mean convergence of the averages () for a Wey systems (properties of d W min (P) were studied in []). Specia Case of Probem. If P 2, show that d W min (P) P. This probem was first stated in [] (set W (P ) := d W min (P) + in the remark after Proposition 5.3). The estimate is known when P = 2, 3, 4 ([2, 40]) and it is open when P = 5. Interestingy, no exampe is known where d min (P) d W min (P), so it is natura to suspect that these two vaues are aways equa. Probem 2. Show that d min (P) = d W min (P). This probem was first stated in []. The identity is known when P = 3 ([2]) and is open when P = 4. Obviousy one has d W min (P) d min(p). Some bounds in the other direction are given in [36]. Mean convergence of the averages () was estabished after a ong series of intermediate resuts; the papers [26, 6, 7, 8, 28, 4, 30, 3, 45] deat with the important case of inear poynomias, and using the machinery introduced in [3], convergence for arbitrary poynomias was finay obtained by B. Host and B. Kra in [32] except for a few cases that were treated by A. Leibman in [35]. Theorem. Let (X, X, µ, T ) be an invertibe measure preserving system, f,..., f L (µ) be functions, and p,..., p be integer poynomias. Then the averages () converge in the mean as. Furthermore, expicit formuas for the imit can be given for specia famiies of poynomias [44, 22, 24, 2, 36], but no such formua is known for genera famiies of poynomias. In most cases, it is sti unknown whether mean convergence can be boosted to pointwise convergence. We mention two particuar cases that are open: Probem 3. Show that the averages f (T n x) f 2 (T 2n x) f 3 (T 3n x), n= converge pointwise. or the averages f (T n x) f 2 (T n2 x), Pointwise convergence of the averages () is known when = [2] and is aso known when = 2 and both poynomias are inear [3] (see aso [9] for an aternate proof). In a other cases the probem is open even for weak mixing systems. Partia resuts that dea with specia casses of transformations can be found in [, 2, 6, 7, 20, 34, 38, 39]..2. Commuting transformations. Throughout this section (X, X, µ) is a probabiity space, T,..., T : X X are commuting, invertibe measure preserving transformations, f,..., f L (µ) are functions, and p,..., p are poynomias with integer coefficients. We start with the foowing resut of V. Bergeson, B. Host, and B. Kra [9]: n=

3 SOME OPE PROBLEMS O MULTIPLE ERGODIC AVERAGES 3 Theorem. For ergodic systems one has the decomposition f 0 T n f... T n f dµ = (n) + e(n) where ((n)) is an -step nisequence and im n= e(n) = 0. A more genera resut that uses poynomia iterates in pace of the inear iterates was proved recenty in [37]. A key ingredient in the proof of the previous theorem is the fact that the factor Z,T is characteristic for convergence of the averages n= f 0 T n f... T n f dµ. When one uses commuting transformations in pace of powers of the same transformation an anaogous property fais, nevertheess, there are no known exampes of muticorreation sequences of commuting transformations that are genuiney different than nisequences. Probem 4. Is it true that one aways has the decomposition f 0 T n f... T n f dµ = (n) + e(n) where ((n)) is an -step nisequence and im n= e(n) = 0? The question is open even when = 2. otice that we make no ergodicity assumptions, so in particuar this probem is open even when = 2 and T 2 = T 2. We move to some probems reated to convergence properties of mutipe ergodic averages. A very natura probem (stated expicity in [8] but was advertised ong before 996 by H. Furstenberg and others) is to extend the mean convergence resut invoving poynomia iterates of a singe transformation to severa commuting transformations: Probem 5. Show that the averages (2) converge in the mean as. n= T p (n) f... T p (n) Mean convergence is known when the transformations T,..., T are powers of the same transformation ([32, 35]), when the poynomias are inear [42] (with aternate proofs given in [43, 3, 29]), and when the poynomias have distinct degrees [5]. Convergence is aso known for genera famiies of poynomias if one imposes very strong ergodicity assumptions on the transformations [33]. See aso [4, 5] where techniques from [3] have been refined and extended, aiming to eventuay hande the case of genera poynomia iterates. Despite such intense efforts, convergence is sti not known for some simpe famiies of poynomias, for instance, when = 2 and p (n) = p 2 (n) = n 2, or when p (n) = n 2, p 2 (n) = n 2 + n. As mentioned previousy, when a transformations are equa, and the poynomias are essentiay distinct, then characteristic factors of the averages (2) can be chosen to have very specia agebraic structure. For genera commuting transformations this is no onger the case; if one chooses p 2 = p = n, T = T 2, and f 2 = f, then the averages (2) do not converge to 0 uness f = f 2 = 0. The same probem persists when two of the poynomias are pairwise dependent, meaning, some non-trivia inear combination of two of the poynomias is constant. But in a other cases, there is no obvious obstruction to having simpe characteristic factors. f

4 4 IKOS FRATZIKIAKIS Probem 6. Suppose that the poynomias p,..., p Z[t] are pairwise independent. Show that there exists d such that the factors Z d,t,..., Z d,t are characteristic factors for the averages (2). This is known to be the case when the poynomias have distinct degrees [5]. But it is not known for some simpe famiies of integer poynomias, for instance, for the famiy {n 3, n 3 + n} or the famiy {n, n 2, n 2 + n}. Even for weak mixing transformations the probem is open: Specia Case of Probem 6. Suppose that the transformations T,..., T : X X are weak mixing and the poynomias p,..., p Z[t] are pairwise independent. Show that im n= T p (n) f... T p (n) f = f dµ... f dµ. When a transformations are equa and the poynomias are in genera position, characteristic factors for the averages (2) turn out to be extremey simpe [22, 24]: Theorem. Suppose that the poynomias p,..., p Z[t] are rationay independent. Then the rationa Kronecker factor K rat (T ) is a characteristic factor for the averages (). It is very ikey that this resut generaizes to the case of severa commuting transformations: Probem 7. Suppose that the poynomias p,..., p Z[t] are rationay independent. Show that the factors K rat (T ),..., K rat (T ) are characteristic factors for the averages (2). This was proved in [5] when = 2 and p (n) = n. In the same artice a somewhat weaker property was proved for a monomias with distinct degrees. We mention aso a cosey reated mutipe recurrence probem: Probem 8. Suppose that the poynomias p,..., p Z[t] are rationay independent and have zero constant term. Show that for every A X and every ε > 0, there exists n such that (3) µ(a T p (n) A T p (n) A) µ(a) + ε. In fact, the set of integers n for which (3) hods is expected to have bounded gaps. The ower bounds are known when a transformations are equa [23] and they are aso known for genera commuting transformations when the poynomias are monomias with distinct degrees [5]. The resut fais if the poynomias are distinct and pairwise dependent; in this case no fixed power of µ(a) works as a ower bound in (3) for every system and set [9]. On the other hand, the assumption that the poynomias are rationay independent is not necessary, for instance, the resut is expected to work for the famiy of poynomias {n, n 2, n 2 + n} (this is known to be the case when a transformations are equa [2]). We remark that Probem 7 is soved, then the conjectured ower bounds of Probem 8 wi foow rather easiy. Regarding pointwise convergence of mutipe ergodic averages of commuting transformations, progress has been extremey scarce. Even when one uses two commuting transformations and inear iterates convergence is not known in genera. The foowing is a we known open probem: Given a measure preserving system (X, X, µ, T ) we define Krat (T ) = d I T d.

5 SOME OPE PROBLEMS O MULTIPLE ERGODIC AVERAGES 5 Probem 9. Let (X, X, µ) be a probabiity space, T, S : X X be commuting invertibe measure preserving transformations, and f, g L (µ) be functions. Show that the averages converge pointwise. f(t n x) g(s n x) n= For a ist of partia resuts that appy to specia casses of transformations see the ist after Probem ot necessariy commuting transformations. A probems in the previous sections were stated for famiies of transformations that commute. It is very ikey that a positive resuts extend to the case where the transformations generate a nipotent group. For instance, we mention a probem from [0]: Probem 0. Let (X, X, µ) be a probabiity space, T,..., T : X X be invertibe measure preserving transformations that generate a nipotent group, and f,..., f L (µ) be functions. Show that the averages converge in L 2 (µ). T n f... T n f n= Convergence is known when = 2 [0] and is open for = 3. The interested reader shoud ook in [0] for a ist of other cosey reated open probems. See aso [?] for a reated mutipe recurrence resut. When one works with arbitrary famiies of invertibe measure preserving transformations the next resut shows that one cannot expect to have simiar convergence resuts: Theorem. Let a, b: Z \ {0} be sequences. Then there exist invertibe Bernoui measure preserving transformations T and S acting on the same probabiity space (X, X, µ) such that for some f, g L (µ) the averages n= T a(n) f S b(n) g dµ diverge; for some A X with µ(a) > 0 we have T a(n) A S b(n) A = for every n. To construct such exampes it suffices to modify exampes of D. Berend (see Ex 7. in [6]) and H. Furstenberg (page 40 in [27]) that cover the case a(n) = b(n) = n (the detais wi appear in [25]). When a(n) = b(n), it is aso known that given any finitey generated sovabe group G of exponentia growth, there exist invertibe measure preserving transformations T, S, with < T, S > G, and such that and the concusion of the previous theorem hods for those T and S. It is interesting that despite such negative news, once one introduces an extra variabe, severa convergence (and very ikey recurrence) resuts can be extended to arbitrary famiies of measure preserving transformations. We mention an exampe from [4]:

6 6 IKOS FRATZIKIAKIS Theorem. Let (X, X, µ) be a probabiity space, T,..., T : X X be invertibe measure preserving transformations, f,..., f L (µ) be functions, p,..., p be essentiay distinct poynomias, and a (0, /d). Then the averages (4) +a converge pointwise as. f (T m+p(n) x)... f (T m+p (n) x) m, n a One can show that the assumption that the poynomias are essentiay distinct is necessary. It was aso shown in [4] that there exists d such that the factors Z d,t,..., Z d,t are characteristic for pointwise convergence of the averages (4). Interestingy, the corresponding mutipe recurrence resut (that woud generaize the poynomia Szemerédi theorem) remains open: Probem. Let (X, X, µ) be a probabiity space, T,..., T : X X be invertibe measure preserving transformations, and p,..., p be distinct poynomias with zero constant term. Show that for every A X with µ(a) > 0 we have for some m, n. 2 µ(a T m p (n) A T m p (n) A) > 0 The assumption that the poynomias are distinct is necessary since as mentioned before there exist (non-commuting) transformations T, S, acting on the same probabiity space (X, X, µ), and a set A X with µ(a) > 0 such that µ(t n A S n A) = 0 for every n. The mutipe recurrence property is known to hod when a the transformations are weak mixing [4], but for genera measure preserving systems even some of the simpest cases are open: Specia Case of Probem. Let (X, X, µ) be a probabiity space and T, S, R: X X be invertibe measure preserving transformations. Show that for every A X with µ(a) > 0 there exist m, n such that µ(a T m A S m n A R m 2n A) > 0. References [] I. Assani. Mutipe recurrence and amost sure convergence for weaky mixing dynamica systems. Israe J. Math. 03 (998), -24. [2] I. Assani. Pointwise convergence of nonconventiona averages. Cooq. Math. 02 (2005), no. 2, [3] T. Austin. On the norm convergence of nonconventiona ergodic averages. Ergodic Theory Dynam. Systems 30 (200), [4] T. Austin. Peasant extensions retaining agebraic structure, I. Preprint, arxiv: v4. [5] T. Austin. Peasant extensions retaining agebraic structure, II. Preprint, arxiv: v3. [6] B. Berend. Joint ergodicity and mixing. J. Anayse Math. 45 (985), [7] B. Berend. Mutipe ergodic theorems. J. Anayse Math. 50 (988), [8] V. Bergeson. Ergodic Ramsey Theory an update, Ergodic Theory of Z d -actions (edited by M. Poicott and K. Schmidt). London Math. Soc. Lecture ote Series 228 (996), 6. 2 This woud impy that given a countabe amenabe group G and arbitrary eements a,..., a G, for every E G that has positive upper density with respect to some Føner sequence in G, there exist g M and m, n such that g, a m+p (n) g,..., a m+p (n) g E.

7 SOME OPE PROBLEMS O MULTIPLE ERGODIC AVERAGES 7 [9] V. Bergeson, B. Host, B. Kra, with an appendix by I. Ruzsa. Mutipe recurrence and nisequences. Inventiones Math. 60 (2005), no. 2, [0] V. Bergeson, A. Leibman. A nipotent Roth theorem. Inventiones Mathematicae 47 (2002), [] V. Bergeson, A. Leibman, E. Lesigne. Compexities of finite famiies of poynomias, Wey systems, and constructions in combinatoria number theory. J. Anayse Math. 03 (2007), [2] J. Bourgain. On the maxima ergodic theorem for certain subsets of the positive integers. Israe J. Math. 6 (988), [3] J. Bourgain. Doube recurrence and amost sure convergence. J. Reine Angew. Math. 404 (990), [4] Q. Chu,. Franzikinakis. Pointwise convergence for cubic and poynomia ergodic averages of noncommuting transformations. To appear in Ergodic Theory Dynam. Systems, arxiv: [5] Q. Chu,. Franzikinakis, B. Host. Ergodic averages of commuting transformations with distinct degree poynomia iterates. To appear in Proc. Lond. Math. Soc., arxiv: [6] J-P. Conze, E. Lesigne. Théorèmes ergodiques pour des mesures diagonaes. Bu. Soc. Math. France 2 (984), no. 2, [7] J-P. Conze, E. Lesigne. Sur un théorème ergodique pour des mesures diagonaes. Probabiités, Pub. Inst. Rech. Math. Rennes, 987-, Univ. Rennes I, Rennes, (988), 3. [8] J-P. Conze, E. Lesigne. Sur un théorème ergodique pour des mesures diagonaes. C. R. Acad. Sci. Paris, Série I 306 (988), [9] C. Demeter. Pointwise convergence of the ergodic biinear Hibert transform. Iinois J. Math. 5 (2007), no. 4, [20] J-M. Derrien, E. Lesigne. Un théorème ergodique poynomia ponctue pour es endomorphismes exacts et es K-systèmes. Ann. Inst. H. Poincaré Probab. Statist. 32 (996), no. 6, [2]. Frantzikinakis. Mutipe ergodic averages for three poynomias and appications. Trans. Amer. Math. Soc. 360 (2008), no. 0, [22]. Frantzikinakis, B. Kra. Poynomia averages converge to the product of integras. Isr. J. Math. 48 (2005), [23]. Frantzikinakis, B. Kra. Convergence of mutipe ergodic averages for some commuting transformations. Ergodic Theory Dynam. Systems 25 (2005), no. 3, [24]. Frantzikinakis, B. Kra. Ergodic averages for independent poynomias and appications. J. London Math. Soc. 74 (2006), no., [25]. Frantzikinakis, E. Lesigne, M. Wierd. Random sequences and pointwise convergence of mutipe ergodic averages. Preprint, arxiv: [26] H. Furstenberg. Ergodic behavior of diagona measures and a theorem of Szemerédi on arithmetic progressions. J. Anayse Math. 3 (977), [27] H. Furstenberg. Recurrence in ergodic theory and combinatoria number theory. Princeton University Press, Princeton, 98. [28] H. Furstenberg, B. Weiss. A mean ergodic theorem for (/) n= f(t n x) g(t n2 x). Convergence in ergodic theory and probabiity (Coumbus, OH, 993), Ohio State Univ. Math. Res. Inst. Pub., 5, de Gruyter, Berin, (996), [29] B. Host. Ergodic seminorms for commuting transformations and appications. S tudia Math. 95 () (2009), [30] B. Host, B. Kra. Convergence of Conze-Lesigne averages. Ergodic Theory Dynam. Systems 2 (200), no. 2, [3] B. Host, B. Kra. on-conventiona ergodic averages and nimanifods. Annas Math. 6 (2005), [32] B. Host, B. Kra. Convergence of poynomia ergodic averages. Isr. J. Math. 49 (2005), 9. [33] M. Johnson. Convergence of poynomia ergodic averages for some commuting transformations. Iinois J. Math. 53 (2009), no. 3, [34] A. Leibman. Pointwise convergence of ergodic averages for poynomia sequences of rotations of a nimanifod. Ergodic Theory Dynam. Systems 25 (2005), no., [35] A. Leibman. Convergence of mutipe ergodic averages aong poynomias of severa variabes. Isr. J. Math. 46 (2005), [36] A. Leibman. Orbit of the diagona in the power of a nimanifod. Trans. Amer. Math. Soc. 362 (200),

8 8 IKOS FRATZIKIAKIS [37] A. Leibman. Mutipe poynomia sequences and nisequences. To appear in Ergodic Theory Dynam. Systems. [38] E. Lesigne. Équations fonctionnees, coupages de produits gauches et théorèmes ergodiques pour mesures diagonaes. Bu. Soc. Math. France 2 (993), no. 3, [39] E. Lesigne, B. Rittaud, T. de a Rue. Weak disjointness of measure preserving dynamica systems. Ergodic Theory Dynam. Systems 23 (2003), [40] D. McCendon. On the maxima Wey compexity of famiies of four poynomias. Preprint, Avaiabe at dmm/papers/fourpoynomias.pdf [4] D. Rudoph. Eigenfunctions of T S and the Conze-Lesigne agebra. Ergodic theory and its connections with harmonic anaysis (Aexandria, 993). London Math. Soc. Lecture ote Ser., 205, Cambridge Univ. Press, Cambridge, (995), [42] T. Tao. orm convergence of mutipe ergodic averages for commuting transformations. Ergodic Theory Dynam. Systems 28 (2008), no. 2, [43] H. Towsner. Convergence of diagona ergodic averages. Ergodic Theory Dynam. Systems 29 (2009), [44] T. Zieger. A nonconventiona ergodic theorem for a nisystem. Ergodic Theory Dynam. Systems 25 (2005), no. 4, [45] T. Zieger. Universa characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20 (2007), (ikos Frantzikinakis) University of Crete, Department of mathematics, Knossos Avenue, Herakion 7409, Greece E-mai address: frantzikinakis@gmai.com