SPTP POLYNOM AND POLYNOM INEQ
SPTP POLYNOM AND POLYNOM INEQ MATH 689
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NOTES FOR MATH 6897TOPICS IN DYNAMICS AND ANALYSIS DAVID KERR September 4 2007 CONTENTS 1 Introduction 1 2 Weak mixing vs compactness for unitary operators 3 3 Basic properties and examples of topological systems 4 4 Combinatorial independence and 1 5 5 Amenable groups 7 6 Topological entropy 9 7 Entropy pairs and IE pairs 11 8 Positive entropy and Li Yorke chaos 13 9 Embedding l f into Md 13 10 The topological Pinsker algebra 14 11 Sequence entropy and nullness 15 12 Rosenthal s ll theorem and tameness 15 13 The Radon Nikodym property and hereditary nonsensitivity 15 References 15 1 INTRODUCTION Dynamics might be described as the analytic or asymptotic study of symmetry As the term dynamics suggests asymptotics may refer to the long term behaviour of a sys tem evolving in time such as the universe In some models as in statistical mechanics dynamics may represent a mechanism for mapping out space rather than time evolution This is the physical motivation for the development of dynamical entropy which provides a means for calculating the entropy of a statistical mechanical system of in nite spatial extent this spatial unboundedness is paradoxically necessary to model phase transitions mathematically In a general abstract sense a dynamical system consists of the action of a group on a space of a certain type by automorphisms ie a homomorphism of a group G into the group AutX of automorphisms of a space X If G is equipped with a topology then one might ask for this homomorphism to be continuous with respect to some topology on AutX In this course we will concentrate on topological and measure preserving systems and our groups will be discrete and countable De nition 11 By a topological dynamical system we mean a pair X G where X is a compact Hausdorff space and G is a countable group acting on X by homeomorphisms We will say that a topological system X G is metrlzable if X is metrizable 1 2 De nition 12 By a measure preserving dynamical system we mean a quadruple X 30 u G where X 30 u is a standard probability space and G is a countable group acting on X 30 u by u preserving bimeasurable transformations We will also speak of G systems On occasion we might need to use a symbol T for the homomorphism representing the action in which case we will write T9 for the image of an element 3 in G Usually however we will simply write sz for the image of a point z E X under T9 and not bother introducing an extra symbol in accord with the notation in the above de nitions We can also describe a system or action as a map G x X a X where sz H sz and 3tm stz for all st E G and z E X In the topological case this map is continuous while in the measure preserving case it is measurable A topological Z system can be alternatively described as a pair X T where T is a homeomorphism from X to itself The action of Z in this case is generated by taking powers of T Similarly a measure preserving Z system can be described as a quadru ple X 30 u T where T is a u preserving bimeasurable transformation of the standard probability space X EelMu lnteger actions are the domain of classical topological and measurable dynamics and all of the basic concepts stem from this setting However new types of phenomena eg rigidity can arise once one moves away from the case G Z In fact the integers are quite special in the sense of being the only nontrivial group which is both free and amenable we will examine these notions later which is important in enabling certain kinds of constructions The main goal of dynamics is to study the asymptotic behaviour of a system as we move outward towards in nity in the group For a Z system with generator T this means looking at the way the iterates T transform the space as n a ioo One basic question is to what degree a system exhibits recurrence One can also ask to what extent the dynamics mixes up the space leading to notions like entropy and weak mixing Our principle aim in this course is to examine recurrence and mixing properties from a systematic perspective based in combinatorial independence and involving ideas rooted in the geometric theory of Banach spaces Taking a perspective which is dual to the usual picture of an action on a space X we can study topological and measurable dynamics in terms of symmetry in Banach spaces or operator algebras by looking at how the dynamics interacts with the linear geometric structure of a suitable space of functions over X We will investigate certain Banach space phenomena in the presence of symmetry and relate these back to what is happening at the level of the space X in order to gain new insight into various dynamical properties These properties will typically involve the notion of independence which originates in probability theory but also appears in combinatorial form in Banach space theory Combinatorial independence manifests itself dynamically in many ways and has long played an important role in the study of topological systems although it has not until now received a uni ed and systematic treatment which is what we aim to provide in this course A recurring theme in both the topological and measurable settings will be the appear ance of dichotomies which separate tame systems from those which exhibit chaotic or random behavior For example we will see how Rosenthal s theorem characterizing those Banach spaces which contain isomorphs of 1 translates dynamically into a tamemixing dichotomy that can be formulated in terms of combinatorial independence Rosenthal s 3 theorem is a prototypical example of a result in Banach space theory which yields the existence of a subspace possessing one of two properties at different extremes This type of dichotomy is essentially local in nature even though in nitedimensional subspaces are involved In the dynamical context where a considerable degree of symmetry is involved it is possible sometimes to leverage local dichotomous behaviour into a global structure theorem This idea underlies Furstenberg s ergodic theoretic approach to Szemeredi the orem on the existence of arithmetic progressions in subsets of natural numbers of positive upper density which we will examine towards the end of the course 2 WEAK MIXING VS COMPACTNESS FOR UNITARY OPERATORS Let 9C be a separable in nitedimensional Hilbert space and U a unitary operator on 9C For a vector 5 6 9C write ZE for the cyclic subspace generated by E ie the closed linear span of the orbit Uni n E Z and 05 for the measure on T whose nth Fourier coefficient is Un 5 Recall that the spectral theorem asserts that there exists a sequence 573311 in EC such that 9C 3 Z n and Hg gtgt agz gtgt and that if 5531 is another sequence in EC with the same properties then 05 N 05 for all n The spectral type try of U is the measure class of 05 We can also record the information captured by the spectral theorem by using 0U along with the multiplicity function MU T a N U Thus up to unitary equivalence a unitary on a separable Hilbert space is a direct sum of operators acting as multiplication by 2 on L2T a for some Borel probability measure a on 339 The measure 0U has an atomic part ad and a continuous part ac These correpond to orthogonal U invariant closed subspaces me and Hep with EC me 63 Hcpct The vectors 5 in Hep are characterized by the relative compactness oftheir orbit Uni n E Z such vectors are called compact vectors The vectors 5 in me on the other hand satisfy the weak mixing condition limH00 2201 lltUi 5 0 Given a Hilbert space 9C its conjugate is the Hilbert space which is the same as 9C as an additive group but with the values of the inner product conjugate to those of 9C for multiplication by a scalar c this means that c5 in E must correspond to 65 in 9C Given a unitary operator U on EC we write U for the unitary operator on E which agrees with U under the identi cation of i and 9C as sets By using the canonical identi cation of 9C X E with the space of Hilbert Schmidt op erators on 9C one can show that the unitary operator U X U on EC X i has a nonzero invariant vector if and only if U has an eigenvector Theorem 21 For a unitary operator U on a separable Hilbert space 9C the following are equivalent 1 11mm 2201 W39s Cgtl 0 for all 5 c e 9c 2 there are positive integers n1 lt n2 lt n3 lt such that limkH00 KUWE 0 for all g 6 9C the only compact vector in 9C is the zero vector U has no eigenvectors 5 U X U has no nonzero invariant vectors 3 4 Theorem 21 extends with a couple of modi cations to unitary representations of arbitrary groups In this case the averaging in condition 1 should be done via the mean on weakly almost periodic functions while condition 4 should assert the nonexistence of nontrivial nitedimensional invariant subspaces References 4 1 3 BASIC PROPERTIES AND EXAMPLES OF TOPOLOGICAL SYSTEMS Many properties of dynamical systems can be categorized as being either of a recur rencetransitivity variety eg minimality transitivity or of a mixing variety eg mixing weak mixing positive entropy These two basic categories are not completely unrelated weak mixing is de ned as transitivity on a product system Minimality is the appropriate notion of irreducibility for a topological system and is de ned as follows The system Y C is a subsystem of the topological system X G if Y is a closed subset of X and the action on Y is the restriction of the action on X De nition 31 A topological system X G is said to be minimal if it has no proper subsystems Proposition 32 A topological system X G is minimal if and only if for every x E X the orbit sx s E G is dense Every topological system has a minimal subsystem by Zorn s lemma However there is no sense in which a general topological system can be decomposed into minimal compo nents this is different from the case of measurepreserving systems Consider for example the left shift action 5 t gt gt st of a countable discrete group G on itself compacti ed with a xed point at 00 This system has a unique minimal subsystem the xed point In particular it cannot be written as a union of minimal systems It does however have a dense orbit De nition 33 A topological system X G is said to be transitive if for all nonempty open sets U V Q X there exists an s E G such that sU O V 7 Q The system is point transitive if it has a dense orbit It is weakly mixing if the product system X x X G is transitive and mixing if for all nonempty open sets U V Q X x X there is a nite set FQGsuchthat sU Vy flforallsEGF Thus X G is weak mixing if for all open sets U1U2 U3U4 Q X there is an s E G such that sU1 Ug 7 Q and sUg U4 7 Q and mixing if for all open sets U1 U2 U3 U4 Q X there is a nite set F Q G such that sU1 U3 7 Q and SU2 U4 7 Q for all s E G F Proposition 34 For topological systems X G with X metrizable transitivity and point transitivity are equivalent De nition 35 A topological system X G is said to be eqvicontinvovs if for every neighbourhood U of the diagonal A x E X there is a neighbourhood V of A such that sV Q U for every 8 E G It is said to be distal if for all distinct xy E X there is a neighbourhood U of A such that sx sy s E G O U Q 5 If X is metrizable then X G is equicontinuous if and only if it is isometric ie there is a compatible metric on X which is G invariant Every topological system has a largest equicontinuous factor and a largest distal factor It is easy to give examples of minimal systems X G with X nite This is less trivial however if X is in nite especially if one wants to arrange for some other properties like positive entropy to hold at the same time An irrational rotation of the circle is a basic and important example of a minimal Z system This system arises as a compacti cation of Z as a group and is equicontinuous All transitive systems arise as a compacti cation of G as a discrete topological space and there is a universal such system with respect to taking factors De nition 36 Let X G and Y C be topological G systems and 7139 X a Y a surjective continuous map such that s7rz 7rsz for all s E G and z E X We refer to 7139 as a factor map or extension X G as an extension of Y C and Y C as a factor of X G The shift action 5 t gt gt st of a countable discrete group G on itself extends to a coninuous action of G on the StoneCech compacti cation BC By the universal property of the StoneCech compacti cation every point transitive system is a factor of 3G G Example 37 The free group F2 acts continuously on its Gromov boundary as follows Let a b be the generators of F2 The Gromov boundary 8F2 is the set of in nite reduced words in a b a l If1 topologized by viewing it as a closed subset of a ba 1b 1N with the product topology It is a Cantor set We let F2 act on X by multiplication on the left reducing as necessary This system is minimal and not distal ls it weakly mixing Example 38 Let G be a countable discrete group and Y a compact Hausdorff space Then G acts continuously on Y0 with the product topology by shifting the index ie Syttea 24960 for all s E G and y tea 6 YO This system is mixing and in particular transitive although it is point transitive only when X is separable Example 39 Let X be the group 01N where addition is modulo 2 with carry over to the right We de ne a homeomorphism T of 01N by Tm z 100 The system X T is called the dyadic adding machine or dyadic odomoter It is minimal and equicontinuous Example 310 Let 9 be an irrational number and de ne the homeomorphism T of T2 by Tzw ezmezgw The system T2T is minimal and distal but not equicontinuous De ning 7r T2 a T by 7rzw 2 yields a factor map of T2T onto its largest equicontinuous factor References 4 18 23 4 COMBINATORIAL INDEPENDENCE AND 61 De nition 41 A collection Ai1 AM i E I of k tuples of subsets of a given set is said to be independent if 1914300 0 for every nite set J Q I and a E 1 k De nition 42 Let X G be a topological system For a tuple A 141142 Ak of subsets of X we say that a set J Q G is an independence set for A if for every nonempty 6 nite subset I Q J and function a I 7 12 k we have HES 11405 7 Q where s lA for a set A Q X refers to the inverse image z E X sz 6 A To realize ll isomorphically in a Banach space V it is enough to produce a bounded sequence of functions whose values on the dual are separated in a uniform way by an independent collection of pairs in Bw as the following proposition demonstrates Proposition 43 Let V be a Banach space Let D0D1 be closed disks in the complex plane with respectiue centres 2021 and common radius r such that r g 20 7 21l6 Let a bounded sequence of uectors in V such that the collection Ai0A1f1 is independent where AM w E Bin E Di Then 21 is 2rilC equiualent to the standard basis of ll where C supiEN Proof We may assume by multiplying the vectors 1 by 21 7 zOlzl 7 20 that 21 7 20 is real and positive lt suf ces to show that for any complex scalars 01 0 we have H 01211 2 2771 21 Writing 0 a ibl we may assume that ELI lail 2 i1 1 Consider a E 0 1 such that 0i is 0 or 1 depending on whether a lt 0 or a 2 0 By independence there exist 8 6 121 AW and t 6 121 1431005 Since refs 7 fit 2 distD0 D1 2 4r when 0i 1 and reft 7 123 2 distD0 D1 2 4r when 0i we have reltafs 7 t2gt 2 2r w Note also that imltZbifi5 7 t2gt E Z lbillimfi3 7 t2l S r Z M S r 21ml i1 i1 i1 i1 Therefore 1 aifis 7 t2gt 7 imltgbifillt3 7 t2gt D Proposition 43 forms part of the proof of the complex version of Rosenthal s ll theorem 19 which was established by Dor in The theorem states that a bounded sequence of vectors in a Banach space has a subsequence which is either weakly Cauchy or equivalent to the standard basis of ll We will discuss it in Section 12 What can we say in the converse direction Given a set S of vectors in a Banach space V which is C equivalent to the standard basis of ll can we nd a collection of pairs of sets in Bw which separate in a uniform way the values of the vectors in S or at least in some subset of S whose relative size depends on C When S is in nite then it follows from 7 Rosenthal s 1 theorem that we can nd an in nite subset of S over which equivalence with the standard 61 basis occurs due to independence In the topological study of entropy and weak mixing however what we need is a quantitative nitedimensional result that we can apply asymptotically as the number of dimensions grows large Given a nite collection 52 Ai1Ak i E I of k tuples of subsets of a given set we can de ne its combinatorial entropy hcz as the number of nonempty intersections of the form mid130 where a E 1 kI If we know something about the entropy hcz can we say something about the possible size of an index subset J C I for which Ai1Ak i E J is independent In the case that k 2 and haw is exponentially large with respect to lIl there will be an index subset of proportional size over which independence occurs as expressed by the following crude form of the Sauer Shelah lemma 20 21 Lemma 44 For eueryb gt 0 there is a c gt 0 such that for alln E N ifS Q 011gt2gt39quotgt satis es 5quot Z elm then there is an I Q 12 n with lIl Z cn and 5 1 01I A similar result holds for general k see lemma 79 and is a consequence of Karpovsky and Milman s generalization of the Sauer Shelah lemma This suggests a combinatorial de nition of entropy for a tuple A1 Ak of subsets of a compact Hausdorff space X with the action of a group G as the growth of the number of nonempty intersections of the form SEF 31405 where a E 1 kF and F is a nite subset of G The problem is to determine exactly how the growth is to be measured ie in what way do we take a limit over nite subsets There is a satisfactory way of doing this if G is amenable which means that G has good averaging properties In our topological framework it is more natural to de ne entropy in a not so strictly combinatorial way using open covers as we will see in Section 6 First however we will need to study amenability to which we turn in the next section References 19 3 6 5 AMENABLE GROUPS Let G be a countable discrete group A mean on 6 0G is a positive unital linear functional 0 6 0G a C Consider the action 04 of G on G given by asft fs 1t for all f E G and s E G The mean a is left inuariantifaozsf 0f for all f e 60 G and s e G De nition 51 We say that G is amenable if 6 0G admits a left invariant mean As a C algebra G is isomorphic to CBG and so by the Riesz representation theorem every left invariant mean on 6 0G corresponds to a regular Borel probability measure on BC Thus G is amenable precisely when there exists a G invariant regular Borel probability measure on BC By the universality of 3G among transitive systems this means that G is amenable precisely when every topological G system admits a G invariant regular Borel probability measure Properties of amenable groups The following Folner characterization of amenability is crucial for many applications Given a nite K Q G and 6 gt 0 we say that a nonempty nite set F Q G is K 6 inuariant if s FKs g Fl 3 17mm Theorem 52 The group G is amenable if and only if for euery nite set K Q G and e gt 0 there exists a nonempty nite setF Q G such that s E F Ks Q Z 176lFl This is equivalent to the existence of a sequence of nonempty nite subsets of G such that limnH00 0 for all s E G The sequence is called a Folner sequence Namioka gave a relatively short proof of the above theorem which produces from an in variant mean approximately invariant functions in 1G whose dual is l G and applies an averaging argument to extract a Folner sequence from level sets of step approximations to these functions To de ne topological entropy in Section 6 we will need a subadditivity result Theo rem 56 below which requires the quasitiling machinery for amenable groups developed by Ornstein and Weiss 16 The key result of Ornstein and Weiss Theorem 55 below uses the Folner characterization of amenability De nition 53 A collection Alha of nonempty nite sets is 6 disjoint if there exist pairwise disjoint sets A Q A with 2 17 6 for all i E I De nition 54 Let G be a group We say that a collection of subsets of G E quasitiles G if there exist a nite set K Q G and a 6 gt 0 such that for every K 6 invariant nite set A C G there is an s disjoint collection Ticij of translates of the T1 satisfying 1A a UTcjlAi 2 17 8 Theorem 55 Let G be a countable amenable group Let 6 gt 0 Then there exists an n E N depending only on e such that ife 6 T1 Q T2 Q Q Tn are subsets of G such that fori 1 n 7 1 we have 83TH mlTZHl for some su iciently small 7 depending oni and Ti then the collection Ti i 1 n e quasitiles G Theorem 56 If 4p is a real ualued function which is de ned on the set of nite subsets of G and satis es 1 0 LpA lt 00 and 0 2 LpA LpB for all A Q B 3 LpAS LpA for all nite A Q G and s E G 4 MAU B S MA wB Z39fz m B 0 then conuerges to some limit b as the set F becomes more and more inuariant in the sense that for euery e gt 0 there exist a nite set K Q G and a 6 gt 0 such that 7 bl lt e for all K 6 inuariant sets F Q G References 13 16 7 6 TOPOLOGICAL ENTROPY Topological entropy was originally formulated for Z actions but the natural general setting is that of actions of amenable groups We will rst treat the classical de nition for Z actions ie7 single homeomorphisms and then extend the de nition to actions of amenable groups using Theorem 56 Let X be a compact metric space and T X 7 X a homeomorphism The topological entropy hmpT7 u of an open cover 11 of X with respect to T is de ned as 1 lim 7 1nNu v T lu v v T Wlu TLHOO 77 where denotes the minimal cardinality ofa subcover The topological entropy ht0pT of T is the supremum of hmpT7 u over all open covers 11 of X We can equivalently de ne entropy in terms of spanning and separated sets Let 1 be a compatible metric on X Given an n E N and 6 gt 07 a set E C X is said to be n6 spanning if for every x E X there is a y E E such that dTizTiy lt 6 for every i 07n 7 17 and n6 separated if for all distinct 71 6 E E X we have dTizTiy gt 6 for some i O7 771 7 1 We write spnn7 8 for the smallest cardinality of an 717 6 spanning set and sepn7 8 for the largest cardinality of an n 6 separated set en 1 1 ht0pT sup lim sup 7 log spnn7 8 sup lim sup 7 log sepn7 8 sgt n7ltxgt n ggt0 7L7gtoo n For every n E N we can de ne a metric dn on X by dnm y max Tim7 Tiy i0n71 so that at the nth stage in the above formulation of entropy we can alternatively speak of s spanning and s separated sets with respect to dn Example 61 The shift on 17 7dZ has entropy log 1 A rotation on the circle is an example of a system with zero entropy In fact Proposition 62 Every homeomomhism of the circle has zero entropy For the remainder of this section X7 G will be a topological system with G a count able amenable group We de ne the topological entropy htopX of X7 G by rst using Theorem 56 to de ne the entropy of a nite open cover 11 of X as the limit of 1 7lnNsu s E 1F 1 as F becomes more and more invariant7 and then de ning ht0pX to be the supremum of these quantities over all nite open covers this was originally introduced in 14 without the subadditivity result In the measurepreserving setting7 the notion of conditional entropy can be used to show that the function 4p to which Theorem 56 is applied to de ne entropy satis es the stronger subadditivite inequality MA U B MAW B S MA MB 10 in which case it is possible to give a simpler proof of the theorem that does not rely on quasitilings see Section 22 of 14 Example 63 The shift on 1 d0 has entropy logd Example 63 illustrates why amenability is the natural context for de ning entropy Consider the points in 1 d as representing states of a statistical mechanical system Each element of G is a site occupied by a particle which can be in one of the at possible states 1 d We wish to determine the mean entropy by taking a limit of the entropy density the logarithmic average of the number of possible states over nite subsets of G along a suitable sequence The appropriate type of nite sets over which to take a limit are Folner sets which play the role of boxes in the case G Z and provide accurate local models for the group structure The structure of the group can effectively be factored out by containing it within approximately invariant nite sets and on nite sets we have a formula for entropy which we can use to take an in nite volume limit In theory we want to average over the whole group but this quantity is ill de ned In the amenable case we can approximately do this by capturing the local structure in nite sets lntuitively positive entropy indicates a certain degree of indeterminism in the system as quanti ed by exponential growth in the number of possible states up to a small obser vational error One of our goals is to show that positive entropy implies local randomness along a subset of powers of T of positive density We can do this by means of combinatorial arguments inspired by problems in Banach space theory We will also describe entropy production at the function level and for this we will need to linearize the measurement of exponential growth so as to be able to invoke tools from the geometric theory of Banach spaces We will now show using the Sauer Shelah lemma how positive entropy produces an isomorph of ll along a positive density subset of the orbit of some function in CX We will reprove this later using a more powerful combinatorial lemma that yields the ll isomorphism by producing independence inside of X The following is Lemma 23 in Lemma 64 For all I gt 0 and 6 gt 0 there exist 0 gt 0 and 6 gt 0 such that for all su iciently large n if E is a subset of the unit ball of along which is 6 separated and lEl Z elm then there are a t 6 711 and a set J Q 12 n for which 1 lJl Z cn and 2 for euery a E 01J there is a u E E such that for allj E J uj2te ifaj1 and W S 25 e if 03 0 Let f E CX We say that J C G is an ll isomorphism set ifsf s E J is equivalent to the standard basis of l ie if the map which sends the standard basis element of l at s E J to sf extends to an isomorphism from l to the closed linear span of sf s E J Proposition 65 Suppose that ht0pX gt 0 Then there is an 1 E CRX such that for euery tempered F lner sequence in G there is an lyisomorphism set J for f such that limnH00 an gt 0 In particular if we are dealing with a Z system X T then the function f has an lyisomorphism set of positiue density References 23 18 4 9 7 ENTROPY PAIRS AND lE PAIRS De nition 71 We call a pair 12 E X x X A2X an entropy pair if whenever U1 and U2 are closed disjoint subsets of X with ml 6 intU1 and 2 6 intU2 the open cover Uf U5 has positive topological entropy More generally we call a tuple 1 1zk E Xk AkX an entropy tuple if whenever U1 Ul are closed pair wise disjoint neighbourhoods of the distinct points in the list m1 zk the open cover Uf Ulc has positive topological entropy De nition 72 We call a pair 1 2 6 X x X an IE pair if for every product neigh bourhood U1 x U2 of 1 2 the pair U1 U2 has an independence set of positive density More generally we call a tuple 1 1 zk E Xk an IE tuple if for every product neighbourhood U1 x x Uk of 1 the tuple U1 Uk has an independence set of positive density We denote the set of lE tuples of length k by lEkX G De nition 73 For a nite tuple A A1 Ak of subsets of X Theorem 56 also applies to the function LpA given by LpAltFgt max F Jl J is an independence set for A and we de ne the independence density A of A as the limit of WHApAltFgt as F becomes more and more invariant A sequence of nonempty nite subsets of G is said to be tempered if for some 0 gt 0 we have lug Fk anl chnl for all n e N Proposition 74 Euery Folner sequence has a tempered subsequence Lindenstrauss established the following pointwise ergodic theorem 12 Theorem 75 Let X 30 a G be a measure preserving system with G amenable and let iFnn be a tempered Folner sequence Then for euery f E L1Xa there is a G inuariant f E L1X a which is equal to itAx if the system is ergodic such that 1 lim 7 1 sm f m W W lt gt lt gt for almost all m E X For a tuple A A1 Ak of subsets of X we write PA for the collection of all independence sets for A We can view PA as a subset of 0 10 by identifying subsets of G with elements in 0 10 via indicator functions The following lemma permits us to convert nitary density statements to in nitary ones The proof makes use of Lindenstrauss s pointwise ergodic theorem Proposition 76 Let X G be a metrizable system Let A A1 Ak be a tuple of subsets of X Let c gt 0 Then the following are equiualent 1 A 2 C 2 for euery e gt 0 there exist a nite set K Q G and a 6 gt 0 such that for euery F E MK6 there is an independence set J for A with l O Fl 2 c7 or eue nite set Q an e gt t ere exist an E e an an in e en 3f ry 39 K G d 0h 39 F MK d 39dp dence set J forA such that JO Fl 2 c7 or euer tern ere oner se uence n n N o t ere is an in e en ence set 4 y p dFl q F 6 G h 39 39 dp d J for A such that limnH00 l fg Z c t ere are a tempere o ner sequence n EN 0 an an in epen ence set or 5 h dF l F G d 39 d d J A such that limnH00 l gg Z c Proof The equivalences 1ltgt2ltgt3 follow from Theorem 56 and the implications 453 are trivial Suppose that 3 holds and let us establish We regard PA as a subset of 0 10 by taking indicator functions It is straightforward to show that there is a G invariant Borel probability measure a on CPA Q 0 10 with ae PA 2 c where 010 is equipped with the shift given by sxt xts for all x 6 010 and st E G Replacing a by a suitable ergodic G invariant Borel probability measure in the ergodic decomposition of a we may assume that a is ergodic Let ELLEN be a tem pered Folner sequence for G The pointwise ergodic theorem 12 Theorem 12 asserts that limnH00 ZSEFT fsx ffdla a ae for every 1 E L1a Setting 1 to be the characteristic function of e O CPA and taking J to be some x satisfying the above equation we get In the case G Z we see from the above theorem that a tuple A A1 Ak has an independence set of positive density if and only if there exists a d gt 0 such that for every M gt 0 there is an interval I in Z with I1 2 M and an independence set J for A included in I for which lJl 2 dlIl The following result provides the combinatorial key for establishing the precise relation ship between entropy tuples and lE tuples Lemma 77 Let k 2 2 and let b gt 0 Then there exists a c gt 0 depending only on k and b such that for euery nite set Z and S Q 0 1 kZ with F5 2 kblzl there exists a W g Z with lwl 2 chl and SW 2 1 kW Lemma 78 Let k 2 2 Let U1 Uk be pairwise disjoint subsets of X and set 11 Uf U Then U U1 Uk has positiue independence density if and only if hCT u gt 0 The following is a consequence of Karpovsky and Milman s generalization of the Sauer Shelah lemma 20 21 8 and can also be deduced from Lemma 77 Lemma 79 Giuen k 2 2 and gt 1 there is a c gt 0 such that for all n E N if S Q 12 k1gt2gtquot39gt satis es 5quot Z 71 then there is an I Q 12 n with in Zen andSl112kI The case lZl 1 of the following lemma appeared in 17 Lemma 710 Let Z be a nite set such that Z O 1 23 There exists a constant c gt 0 depending only on lZl such that for alln E N ifS Q ZU1 21gt2gt39quotgt is such that ms S a Zo31gt2gt n is bijectiue where rn ZU1 212w a Zo31gt2gt n conuerts the coordinate ualues 1 and 2 to 3 then there is some I Q 12n with lIl Z cn and either 5 1 2 ZU 11 orSlI 2 ZU 21 8 POSITIVE ENTROPY AND LI YORKE CHAOS The notion of Li Yorke chaos was introduced in 2 and is motivated by the study of interval dynamics in 11 Let X T be a metrizable topological Z system and let 1 be a compatible metric on X We say that a pair 12 E X x X is a Li Yorke pair with modulus 6 if lim sup pT z1T m2 6 gt 0 and liminfpT m1T z2 0 naoo 4 00 We say that a set Z Q X is scrambled if all nondiagonal pairs of points in Z are Li Yorke The system X T is said to be Li Yorke chaotic if X contains an uncountable scrambled set For a subset P of 92 we say that a nite subset J C G has positive density with respect to P if there exists a K Q G with positive density such that K 7 K O J 7 J 0 and K J E P We say that a subset J Q G has positive density with respect to P if every nite subset of J has positive density with respect to P Lemma 81 Let P be a hereditary closed shift invariant subset of 01Z with positive density Then there exists a J Q 220 with positive density which also has positive density with respect to P Theorem 82 Let k 2 2 and let 1 1zk be an lE tuple in Xk AkX For each 1 g j g k let Aj be a neighbourhood of xi Then there exist a 6 gt 0 and a Cantor set Zj Q Aj for eachj 1 k such that the following hold 1 every nonempty nite tuple of points in Z U Zj is an lE tuple 2 for allm E N distinct yl ym E Z and 31 yn E Z we have n l l gflgag p up 0 Corollary 83 If htopT gt 0 then X T is Li Yorke chaotic Corollary 84 The set lE2X T of IE pairs does not have isolated points 9 EMBEDDING l f INTO Md We write Md for the C algebra of d x 1 matrices over the complex numbers and Tr for the unnormalized trace on Md ie Trajj 2211 a Recall that all of our Banach spaces are over the complex numbers unless otherwise indicated We will need the generalized Horn inequality Associated to a matrix a E Md are its s numbers 31a 2 52a 2 2 sda which are the eigenvalues of la a ia12 with multiplicity Theorem 91 For a1 aq E Md we have k Trltla1aqlgt Zsilta1gtsltaqgt The type 2 constant T2V of a Banach space V is de ned as the in mum of the positive numbers T such that for all m E N and U1 um E V 1 m 2 m 27 Z Zam T2 HWHZ i391 515mi1 i1 Notice that T2llf 2 n as can be seen by considering the standard basis U1 un Lemma 92 We have T2Md Z cxlogd for d 2 2 where c gt 0 is a universal constant Theorem 93 Let E be a n dimensional subspace of Md which is A isomorphic to 6 Then n g a2 logd where a gt 0 is a universal constant References 22 15 5 10 THE TOPOLOGICAL PINSKER ALGEBRA Let X G be a topological system This gives rise to an action 04 of G on C X by automorphisms de ned by a9fz fs 1m for all f E CX z E X and s E G This is an example of a C dynamical system For the remainder of this section G will be assumed to be amenable We denote by FinCX the collection of nite subsets of CX As before the 0 algebra of d x 1 matrices over C is written Md For each 9 Q P nX and 6 gt 0 we denote by CMAQ 6 contractive matricial approximation the collection oftriples 4p w d where d is a positive integer and 4p CX a Md and w Md a CX are contractive linear maps such that W 0 WV 7 ill lt 5 for all f E Q We then set rmltn6gt infw wad e CMAlt96gt Considering our action 04 from above we now de ne 1 hma 96 ling FEE111p6 logrmozs 2 s E F6 hma Q sup hm 04 Q 6 6gt0 hma Q sup hma QeFinCX Lemma 101 Let Q f1 be a nite subset of CX and suppose that the linear map 39y l gt X sending the ith standard basis element of l to m for each i 1 n is an isomorphism Let 6 gt 0 be such that 6 lt H39yilll l Then 10grm97 5 Z WHVH ZUWTIHTI 32 where a gt 0 is a universal constant 15 Proof Let Ap1l1d E CMAQ6 For any linear combination Zea of the elements we have 2 ZWWWKZWMMllltwwgtltZcmgtH wzwwzcmw swim llvltZcmgtH and so 2 17 639y 1 Since p is contractive it follows that the composition Lpo y is a y39y 1 1 76 1 isomorphism onto its image in Md The desired conclusion now follows from Theorem 9 3 Lemma 102 For euery 6 gt 0 there exist 0 gt 0 and e gt 0 such that for euery com pact Hausdor space Y and nite subset 9 of the unit ball of CY of su iciently large cardinality if the linear map 39y l gt CY sending the standard basis of l to 9 is an isomorphism with y 1H 1 Z 6 then there exist closed disks B1 B2 Q C of diameter at most 86 with distB1B2 Z 6 and an I Q 9 with I Z c9 such that the collection f71B1f 1B2 f E I is independent Theorem 103 Let f E CX Then the following are equiualent 1 f if J3XG hmaf gt 0 there is an IE pair xy E X X X with 7 MAX gt 0 for euery tempered Folner sequence in G there is an ll isomorphism set J for f such that limnH00 Fn gt 0 2 3 4 5 Example 104 The StoneCech compacti cation 32 provides an example of a compact Hausdorff space which admits a G action with in nite 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