Dynamical Morse entropy - IHESgromov/wp-content/uploads/2018/08/...Dynamical Morse entropy M´elanie...

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Dynamical Morse entropy elanie Bertelson ** & Misha Gromov April 30, 2010 * epartement de Math´ ematiques UCL 2, Chemin du Cyclotron 1348 Louvain-la-Neuve Belgique [email protected] IH ´ ES 35, route de Chartres 91440 Bures-sur-Yvette France [email protected] 1 Introduction Consider a ”crystal”, that is, the standard lattice Γ = Z 3 in R 3 with identical ”molecules” positioned at all sites (points) γ in Γ. Denote by M the configuration space of such a molecule which is assumed to be a smooth finite dimensional manifold and let X = M Γ be the configuration space of the crystal, that is, the infinite product of Γ copies of M . Suppose adjacent molecules interact via a potential (energy) which is, by definition, a smooth function of two variables, say f : M × M R. Then the total energy of the crystal could be thought of as the (infinite !) sum of copies of f over all adjacent pairs of sites (γ,γ ), called edges of Γ (with six edges at each site) : F (x =(x γ )) = edges(γ,γ ) f (x γ ,x γ ). (*) Such an F is clearly almost everywhere infinite for non-trivial f but its gradi- ent (differential) is obviously well defined and finite at all points x in X. Thus one may speak of the critical points of F , also called the stationary states of the crystal. Observe that the set S of stationary states make a closed subset in X invariant under the obvious (shift) action of Γ on X. The basic question is that of evaluating the entropy of the dynamical system (S, Γ) in terms of the topology * Research supported by a FNRS Charg´ ee de Recherches contract 1

Transcript of Dynamical Morse entropy - IHESgromov/wp-content/uploads/2018/08/...Dynamical Morse entropy M´elanie...

Dynamical Morse entropy

Melanie Bertelson!!& Misha Gromov†

April 30, 2010

! Departement de MathematiquesUCL

2, Chemin du Cyclotron1348 Louvain-la-Neuve

[email protected]

† IHES35, route de Chartres

91440 Bures-sur-YvetteFrance

[email protected]

1 Introduction

Consider a ”crystal”, that is, the standard lattice ! = Z3 in R3 with identical”molecules” positioned at all sites (points) ! in !. Denote by M the configurationspace of such a molecule which is assumed to be a smooth finite dimensionalmanifold and let X = M! be the configuration space of the crystal, that is, theinfinite product of ! copies of M . Suppose adjacent molecules interact via apotential (energy) which is, by definition, a smooth function of two variables, sayf : M !M " R. Then the total energy of the crystal could be thought of as the(infinite !) sum of copies of f over all adjacent pairs of sites (!, !"), called edgesof ! (with six edges at each site) :

F (x = (x!)) =!

edges(!,!!)

f(x! , x!!). (#)

Such an F is clearly almost everywhere infinite for non-trivial f but its gradi-ent (di"erential) is obviously well defined and finite at all points x in X. Thusone may speak of the critical points of F , also called the stationary states of thecrystal. Observe that the set S of stationary states make a closed subset in Xinvariant under the obvious (shift) action of ! on X. The basic question is thatof evaluating the entropy of the dynamical system (S, !) in terms of the topology

!Research supported by a FNRS Chargee de Recherches contract

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of M and/or some generic features of f . One still does not have a satisfactorycriterion for non-vanishing of this entropy except for a few specific cases, such asthe discretized geodesic flow on a Riemannian manifold for instance, but one cangive a lower bound on the asymptotic distribution of the critical values of F asfollows.

Exhaust ! with some standard subsets #i, e.g. by concentric cubes of edge size2i, and let Fi denote the ”restriction” of F to #i, that is, the sum of the termsin (#) corresponding to edges in #i. This sum is regarded as a function on M"i ,call it Fi : M"i " R. It is further normalized by letting F "i = 1/ card(#i) Fi.The functions F "i take values in a fixed interval, namely in [f$ = 6 inf(f), f+ =6 sup(f)]. We count the number #i(I) of critical values in each subinterval I of[f$, f+], and set

crii(I) =1

card(#i)log #i(I).

The purpose of this paper is to provide a Morse theoretic lower bound for lim inf crii(I),i " %, in terms of a certain (strictly positive !) concave function (entropy) on[f$, f+] capturing the homological behaviour of functions F "i for i "%.

Aknowledgements We wish to thank the referee for having noticed some mis-takes in the text.

2 Framework

Consider a countable group ! endowed with a left-invariant metric d : ! ! ! "R+. Given a finite subset # & !, its cardinality is denoted by |#|. The set offinite subsets of ! is denoted by B(!). The distance d is extended to a mapd : B(!)!B(!) " R+ (not a distance) as follows :

d(#,#") = inf{d(!, !"); ! ' #, !" ' #"}.

Given a nonnegative number N , the N -boundary and the N -interior of # ' B(!)are the sets

"N# = {! ' !; d(!,#), d(!,!$ #) ( N},intN# =# $ "N#.

When reference to N is clear the set intN/2# will be denoted by "#. Given #o and#, two finite subsets of !, we denote their amenability ratio by #(#,#o), that is

#(#,#o) =|"Do#||#| ,

where Do = sup{d(!, !"); !, !" ' #o} is the diameter of #o. Let us recall that acountable group is said to be amenable if it admits an amenable sequence (#i),i.e. an increasing sequence of finite subsets exhausting ! such that for any non-negative number N ,

limi#$

|"N#i||#i|

= 0.

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Let X be a compact topological space endowed with a !-action $ : ! !X "X : (!, x) )" !x.

Let fo : X " R be any continuous function with fo(X) = [0, 1]. For # in B(!),we define the average of fo along # to be the function

f"(x) =1|#|

!

!%"

fo(!&1x).

3 Product-like actions

We will impose on the group action $ a restrictive assumption of homologicalnature expressing abundance of multiplicative structure. It will ensure that thehomological measure defined in the next section will have a well-defined exponen-tial growth. Its statement requires the introduction of some elements of notation.

Let F be a field and let H"(X;F ) denote the singular cohomology of X withcoe$cients in F . Given a finite-dimensional subalgebra A & H"(X;F ) and a finitesubset # of !, we denote by A" the (finite-dimensional) subalgebra of H"(X;F )generated by the translates of A along #, i.e.

A" = Alg# $

!%!

!!A%,

where !! = (!&1)! denotes the induced (left) action of ! on H"(X;F ).

Assumption 3.1 There exists a nontrivial subalgebra A & H"(X;F ) for whichany finite-dimensional subalgebra A & A admits a number N = N(A) * 0 suchthat if #,#" ' B(!) satisfy d(#,#") > N(A), then the cup product map is injective

A" +A"! %" A"'"! : a+ a" )" a , a". (!)

Remark 3.2 Assumption 3.1 The word nontrivial in the statement above shouldbe given the meaning that the algebra A contains some nonzero finite-dimensionalalgebra.

When this assumption is satisfied, the action $ is said to be a product-like action,in reference to the following example.

Example 3.3 (Products) Let M be a manifold. Consider X = M!, the infiniteproduct of ! copies of M , or equivalently, the set of maps

! " M : ! " x! ,

with the topology of pointwise convergence (or product topology). Assumption 3.1holds. Indeed, an algebra A satisfying the condition (!) is the direct limit of the

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direct system of subalgebras described hereafter. To each finite subset # & ! isassociated a finite-dimensional subalgebra A(#) of H"(M!;F ) :

A(#) = p!"(H"(M";F )),

where p" : M! " M" is the canonical projection. When #" & #, there is a mapp","! : M" " M"!

and hence a pullback i","! : A(#") " A(#). The algebra

A = lim$""%B(!)

A(#)

is a subalgebra of H"(M!;F ) that satisfies (!). Indeed, let A & A be a finite-dimensional subalgebra. There exists a finite subset #o & ! such that A & A(#o).Given #, the space A" is contained in A(# · #o) and, as the Kunneth formulaimplies, it su$ces to use N(A) = diam(#o · #&1

o ).

A class of examples of !-spaces not of the product type, but enjoying the product-like property is described below in Section 12.

Remark 3.4 Assumption 3.1 is not satisfied when X is a manifold, or whenH"(X;F ) has finite rank.

4 Homological measure of thickened level sets

We will define homological invariants associated to a continuous function fo on X.They can be interpreted, roughly speaking, as a measure of the amount of coho-mology supported in the various thickened level sets of the averages of fo over thefinite subsets of !, and therefore could be called homological measures of slices. IfX was a manifold and if fo wasa smooth function, these invariants would provide ameasure for the number of ”homologically-detectable” critical points of the variousfunctions f" located in the various thickened level sets of f" (cf. Section 8). Thereal purpose is to consider the exponential growth of this invariant as the finite sub-set becomes large. The resulting object will depend upon two variables : the leveland the normalized degree in cohomology. It is called hereafter the homologicalentropy of the function fo, in analogy with the traditional entropy of an observ-able ([4]). The entropy is well-defined provided these invariants satisfy certainproperties. Classically these properties are submultiplicativity and !-invariance.In contrast, the homological measure is invariant as well (Lemma 5.2), but su-permultiplicative (Lemma 5.1). To define entropy in this situation necessitatesthe introduction of an additional assumption on the group, called here tileability(cf. Section 6).

Notation 4.1 If a is a cohomology class and if O is an open subset of X, theexpression supp a & O (”a is supported in O”), means that for some open setO" such that X = O - O", the restriction of a to O" vanishes. Observe that ifsupp a & O and supp b & U then supp a , b & O . U .

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Given A & A a finite-dimensional subalgebra, an open set O, a non-negativenumber &, a positive number ' and a finite subset # & !, we consider the subal-gebras

H"Ae!

(O) = {a ' Ae"; supp a & O},H#,$

Ae!(O) = {a ' Ae"; supp a & O & (&$ ')|#| < deg a < (& + ')|#|}.

Recall the convention that "# denotes intN/2 #, where N = N(A) is the numberassociated to A & A from Assumption 3.1. The inequalities involving the degreeof a have to be verified by each component of pure degree. Finally, the opensets considered hereafter will be sublevel, superlevel or thickened level sets of thefunction f", typically :

O = f&1" ($%, c + () or f&1

" (c$ (,+%) or f&1" (c$ (, c + (),

for some c ' [0, 1] and ( > 0. Then consider the map

)#,$A,",c,% : H#,$

Ae!(f&1

" ($%, c + ()) "

Hom&

H"Ae!

(f&1" (c$ (,+%)), H"

Ae!(f&1

" (c$ (, c + ())'

()#,$

A,",c,%(a))(b) = a , b.

Its rank is denoted hereafter by

b#,$A,"(c, () = rank)#,$

A,",c,%.

Definition 4.2 b#,$A,"(c, () is called the (&$ ', &+ ')-th homological measure of the

thickened level f&1" (c$ (, c + () with respect to A".

5 Properties of the homological measure

We prove in this section the two properties – supermultiplicativity and !-invariance– necessary to obtain a well-defined homological entropy.

Lemma 5.1 The map B(!) " R : # )" b#,$A,"(c, () is supermultiplicative. More

generally, let #,#" ' B(!) be disjoint finite subsets, let c, c" ' [0, 1] and let &, &" * 0,then

b&#+(1&&)#!,$A,"'"! (#c + (1$ #)c", () * b#,$

A,"(c, () · b#!,$A,"!(c", (), (1)

where # = |"||"'"!| and thus 1$ # = |"!|

|"'"!| .

Proof. The argument relies on the few simple observations listed below :

- Let # and #" be disjoint finite subsets of !, then

f"'"! = |"||"'"!|f" + |"!|

|"'"!|f"! = #f" + (1$ #)f"! .

Thus f&1"'"!(#I + (1$ #)I ") / f&1

" (I) . f&1"! (I ") for intervals I and I ".

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- If a class a is supported in f&1" (I) and if a class a" is supported in f&1

"! (I "),then the class a,a" is supported in f&1

" (I).f&1"! (I ") & f&1

"'"!(#I +(1$#)I ").

- If # and #" are disjoint then the distance between "# and "#" is greater thanN and "# - "#" & !# - #". Therefore Assumption 3.1 provides us with aninjective map

Ae" +Ae"! " A!"'"! .

This explains the choice of "# instead of #.

- The degree of a , a" if the sum of the degree of a and that of a". Thus

a ' H#,$Ae!

a" ' H#!,$Ae!!

*0 a , a" ' H&#+(1&&)#!,$

A!!"!!.

Combining the previous observations we obtain an injection :

%I,I! : H#,$Ae!

(f&1" (I))+H#!,$

Ae!!(f&1

"! (I ")) " H&#+(1&&)#!,$A!!"!!

(f&1"'"!(#I + (1$ #)I ")).

Now consider the following sequence of maps. We will abbreviate #c + (1 $ #)c"to c and #& + (1$ #)&" to &.

H#,$Ae!

(f&1" ($%, c + ())

+H#!,$

Ae!!(f&1

"! ($%, c" + ())1

Hom&

H"Ae!

(f&1" (c$ (,+%)), H"

Ae!(f&1

" (c$ (, c + ())' +

Hom&

H"Ae!!

(f&1"! (c" $ (,+%)), H"

Ae!!(f&1

"! (c" $ (, c" + ())'

1

Hom&

H"Ae!

(f&1" (c$ (,+%))+H"

Ae!!(f&1

"! (c" $ (,+%)),

H"Ae!

(f&1" (c$ (, c + ())+H"

Ae!!(f&1

"! (c" $ (, c" + ())'

1

Hom&

H"A!!"!!

(f&1"'"!(c$ (,+%)), H"

A!!"!!(f&1

"'"!(c$ (, c + ())'

.

The first arrow stands for the map )#,$A,",c,% + )#!,$

A,"!,c!,%. The second arrow is aclassical isomorphism, indeed,

Hom(A, B)+Hom(C, D) 2 Hom(A+ C, B +D)

for finite-dimensional vector spaces A, B, C, D. The third one is the injection in-duced by a choice of complementary subspaces to the images of the maps %(c&%,+$),(c!&%,+$)

and %(c&%,c+%),(c!&%,c!+%) respectively. We will denote the composition of secondand third map by &. There is also another sequence obtained from composing

% =% (&$,c+%),(&$,c!+%)

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with)&#+(1&&)#!,$

A,"'"!,&c+(1&&)c!,%.

These two sequences commute :

& 3,)#,$

A,",c,% + )#!,$A,"!,c!,%

-= )&#+(1&&)#!,$

A,"'"!,&c+(1&&)c!,% 3%.

Since & and % are both injective, this implies that

rank()#,$

A,",c,% + )#!,$A,"!,c!,%

)( rank)&#+(1&&)#!,$

A,"'"!,&c+(1&&)c!,%

Lemma 5.2 The map

B(!) " R : # " b#,$A,"(c, ()

is !-invariant.

Proof. The proof follows from the simple facts stated below. Let # ' B(!), let!o ' ! and let I & [0, 1] be any interval. Then

- f&1!o"(I) = !of

&1" (I).

- A!o" = (!o)!A".

- .!o# = !o"#.

- (!o)! induces an isomorphism between H#,$Ae!

(f&1" (I)) and H#,$

!o#Ae!(!of

&1" (I)).

6 Superadditive Ornstein-Weiss Lemma for tilea-ble groups

The &-th Betti number entropy of fo will be defined from the exponential growth,with respect to the index i, of the sequence (b#,$

A,"i(c, ())i(1, where #i is an

amenable sequence in !, that is to say, from the limit

limi#$

ln(b#,$A,"i

(c, ())|#i|

,

when it exists. Lemma 5.1 implies that the map # )" ln(b#,$A,"(c, ()) is superadditive

on disjoint sets, while the Ornstein-Weiss lemma [3] provides convergence of thesequence of averages h(#i)/|#i| under the hypotheses that the map h : B(!) "R+ is !-invariante and subadditive. The proof of the Ornstein-Weiss Lemmarequires the construction of *-quasi-tilings that any amenable group admits. Incontrast, a proof of the superadditive version of this lemma seems to necessitatethe construction of disjoint such tilings which might not exist in general (althoughwe do not know of any counterexample). Whence the following definition.

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Definition 6.1 An amenable group ! is said to be tileable if it admits a tilingamenable sequence, that is, an amenable sequence (#i) such that given * > 0 anda subsequence (#in) of (#i), there exists a finite subsequence of (#in), denoted#1, . . . ,#s, such that any finite subset # with su!ciently large amenablility ratios#(#,#j), j = 1, . . . , s can be disjointly *-tiled by translates of the #j’s, i.e. thereexists center !j,k, 1 ( j ( s, 1 ( k ( rj in ! such that

- !j,k#j & #,

- !j,k#j . !j!,k!#j! = 4 for (j, k) 5= (j", k"),

- | -j,k !j,k#j | * (1$ *)|#|.

Examples 6.2 Weiss introduces in [5] the notion of monotileable amenable groups.Those are groups admitting an amenable sequence (#i) consisting of monotiles.This means that for each index i there exists a set Ci & ! for which the varioustranslates #ic of #i along Ci form a partition of !. Such groups belong to theclass of tileable amenable groups. Moreover, Weiss proves that any residually finiteamenable group is monotileable, implying that the following amenable groups arealso tileable.

- Abelian and solvable groups.

- Amenable linear groups, i.e. linear groups not containing F2 as a subgroup.

- Grigorchuk’s groups of intermediate growth.1

Lemma 6.3 (Superadditive Ornstein-Weiss lemma) Let ! be a tileable amenablegroup. Let h be a nonnegative function defined on B(!) and satisfying the followingtwo conditions :

- superadditivity : h(# - #") * h(#) + h(#") for disjoint subsets # and #",

- !-invariance : h(!#) = h(#) for any ! ' !.

Then, given a tiling amenable sequence (#i), the following limit exists

limi#$

h(#i)|#i|

.

Remark 6.4 Observe that under the hypotheses of the previous lemma, the limitis independent of the choice of a tiling amenable sequence in !.

Proof. Let * > 0 and let (#i) be a tiling amenable sequence. Extract a sub-sequence #i1 , . . . ,#is with which we can *-tile any element #i of the initial se-quence with su$ciently large index. Suppose also that if h+ stands for thelimsup of the sequence h(#i)/|#i|, then h(#ij )/|#ij | * h+ $ * for all j. Let

1Grigorchuk, Rotislav I., Degrees of growth of finitely generated groups and the theory ofinvariant means. Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 939–985.

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!j,k, 1 ( j ( s, 1 ( k ( rj denote the centers of a disjoint *-tiling of #i bytranslates of the #ij ’s and let #"i = -j,k !j,k#ij . Then

1|#i|

h(#i) * 1|#i|

(h(#"i) + h(#i $ #"i)

)* 1

|#i|h(#"i)

* 1|#i|

/j,k h(#ij ) *

1|#i|

/j,k(h+ $ *)|#ij |

* 1|#i|

(h+ $ *)(1$ *)|#i| = (h+ $ *)(1$ *).

Hencelim infi#$

1|#i|

h(#i) * (h+ $ *)(1$ *).

Since this holds for arbitrary *, the limit limi#$ h(#i)/|#i| exists.

7 Homological entropy of functions

Let (#i)i(1 be a tiling amenable sequence in the tileable amenable group ! andconsider the exponential growth of the sequence b#,$

A,"i(c, () :

b#,$A (c, () = lim

i#$

1|#i|

ln0b#,$A,"i

(c, ()1. (2)

As implied by Lemma 5.1, Lemma 5.2 and Lemma 6.3, this limit indeed exists.Observing that the function b#,$

A (c, () is increasing in ' and (, we let ( and 'approach 0 :

b#A(c) = lim

$#0lim%#0

b#,$A (c, ().

Independence on A is obtained by considering the supremum over all possiblechoices of a finite-dimensional subalgebra A of A :

b#(c) = sup{b#A(c);A & A & dim A < %}.

Definition 7.1 The function b# : [0, 1] " R : c )" b#(c) is called the &-th Bettinumber entropy of fo.

Remark 7.2 One may also define the sum of the Betti number entropy of fo bythe same process except that the cohomological degree is not restricted. The twofunctions are related as follows :

b(c) = sup#

b#(c).

(This is a consequence of the general fact that the exponential growth of the sum oftwo sequences coincides with the exponential growth of the maximum sequence.)

Remark 7.3 The condition that ! be tileable is only used to guarantee existenceof the limit (2).

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8 Relation with classical Morse theory

This section is devoted to showing how, in the setting of a manifold M endowedwith a Morse function f : M " R, the sum of the Betti number entropy essentiallycoincides with the sum of the Betti numbers of M and provides a lower bound forthe number of critical points of f (cf. Proposition 8.1).

Let M be a connected, closed, oriented smooth manifold. Consider a Morsefunction f on M . Let F be a field. We recall that if O is some subset of M andif a is a cohomology class in H"(M ;F ), the expression supp a & O means thata|O! = 0 for some open subset O" containing M $ intO. We denote by Critc(f)the set of critical points of f at level c. Define

H"(O) =2

a ' H"(M ;F ); supp a & O3

,

)c,% : H"(f&1($%, c + ()) " Hom(H"(f&1(c$ (,+%)), H"(f&1(c$ (, c + ())

),

a )"0b )" a , b

1,

b(c, () = rank)c,%,

b(c) = lim%#0 b(c, ().

We might sometimes denote b(c, () by b(c $ (, c + () or consider b(I) when I issome interval.

Proposition 8.1

(a)!

c%Rb(c) = SB(M).

(b) b(c) ( Critc(f).

Proof.(a) The main ingredient is the specific version of Poincare duality mentioned belowin Section 10.1. Indeed, if a ' H"(M ;F ), define

ca = inf{c; supp a & f&1($%, c)}.

Since for all ( > 0 the restriction of a to f&1(ca $ (,+%) does not vanish, thereexists a class b with supp b & f&1(ca $ (,+%) such that a , b 5= 0 (cf. Proposi-tion 10.2). Hence a provides a contribution to b(ca). Here follows a more preciseargument taking into account the following di$culty : there might exist two classesthat have same ca and are independent, but who do not generate a 2-dimensionalspace of classes with same ca.

Decompose the range I of fo into intervals as follows :

I &K4

k=1

(Ik = [ak, ak+1)) ak < ak+1.

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If J & R, let rJ : H"(M ;F ) " H"(f&1(J);F ) denote the restriction map. Thenconsider the following increasing sequence of subspaces

{0} = Ker rI1'...'IK & ... & Ker rIK$1'IK & Ker rIK & H"(M ;F ).

Choose a corresponding sequence of spaces V1, ..., VK such that

Ker rIk'...'IK 6 Vk = Ker rIk+1'...'IK k = 1, ...,K.

(For k = K, we mean Ker rIK 6 VK = H"(M ;F ).) If 0 5= a ' Vk then ca ' Ik.Hence b(Ik) = rankVk and

K!

k=1

b(Ik) = SB(M). (3)

This is true for arbitrarily fine subdivisions of I. Now observe that for each c,either b(c) = 0, in which case b(c $ (, c + () = 0 for all su$ciently small ( > 0,or b(c $ (, c + () 5= 0 for all ( > 0. Relation (3) implies that there are finitelymany numbers c with b(c) 5= 0. So

/k b(Ik) is constant, equal to

/c b(c), for all

su$ciently fine subdivisions of I.

(b) Given a ' H"(M ;F ), ca must be a critical value, otherwise we would be ableto move f&1($%, c+ () below level c$ ( by an ambient isotopy, disjointifying thesupports of the classes a and b. In consequence, a , b would vanish. This aloneimplies (b) when b(c) ( 1 for all c. We will argue that if b(c) = 2 then f cannothave a single critical point at level c (the general case can be handled in a similarway). Suppose on the contrary that {x} = Critc(f). Let x1, ..., xm be coordinateson M , centered at x, for which f has the canonical form

f(x) = $x21 $ ...$ x2

n + x2n+1 + ... + x2

m.

Let a1 and a2 be two independent classes with ca1 = ca2 = c. Consider piecewisesmooth cycles #1 and #2 representing their Poincare dual homology classes. Wewill make the following assumptions on #i, i = 1, 2 :

- #i is supported in f&1($%, c],

- #i . f&1(c) = {x},

- x is a regular value of (each of the simplices composing) #i,

- #i intersects the local unstable manifold Wu(x) = {x;x1 = ... = xn = 0} ofx transversely at x.

It is long but not di$cult to verify that these hypotheses are not restrictive.

Now, we will show that the degree of #i must equal the index n of x. Thedegree of #i can certainly not exceed n, otherwise #i would not be supported inf&1($%, c]. If the degree of #i was less than n, one could slide #i down the stable

11

manifold of x (in a direction transverse to that of Tx#i) below level c.

Then, one can subdivide all the simplices of #i containing x in such a waythat x lies in the interior of each simplex to which it belongs and that each suchsimplex can be isotoped to a fixed simplex that coincides with the stable manifoldWs(x) of x in a neighborhood of x. Thus,

#i = f0i +0

i +!

j(1

f ji +j

i ,

where f0i , f j

i ' F , where +0i is a piece of Ws(x) containing x and where +j

i avoidsx. It follows that f0

1 #2 $ f02 #1 vanishes near x, hence that cf0

1 a2&f02 a1 < c. So a1

and a2 do not generate a space contributing to b(c), a contradiction.

Remark 8.2 The previous lemma implies the Morse theoretic lower bound an-nounced in the introduction. Referring to the notation used therein, one observesthat the previous proof implies in particular that if I & R is some interval, thenCritI(F "i ) * bA,"i(I), where A 2 H!(M ;F ). Hence

lim infi#$

crii(I) * bA(I).

(The function F "i defined in the introduction does not quite coincide with thefunction f"i defined in Section 2, but the di"erence will not a"ect the asymptoticbehavior of the objects considered here). Moreover, as proved later on, the functionbA(I) is concave and strictly positive.

9 Concavity of the entropy

The above-defined function b : R+ ! [0, 1] " R is concave. This follows mainlyfrom Lemma 5.1, with a slight help from the following fact.

Lemma 9.1 The function b is upper semi-continuous.

Proof. Let (&k, ck) be a sequence converging to some pair (&, c) in R+![0, 1]. Sinceb#,$A,"(c, () is increasing with respect to both intervals (&$', &+') and (c$(, c+(),

b#,$A,"(c, () * b

#k, !2

A," (ck, %2 )

for su$ciently large k and for all A and #. Hence b#(c) * b#k(ck). Thus b#(c) *lim supk#$ b#k(ck).

Proposition 9.2 The function b is concave. That is to say, for any &, &" ' R+,any c, c" ' [0, 1], and any # ' [0, 1],

b&#+(1&&)#!(#c + (1$ #)c") * # b#(c) + (1$ #) b#!(c"). (4)

12

Proof. Let (#i) be an amenable sequence. For each i, let #"i = !i · #i be disjointfrom #i. Then the sequences (#"i) and (#i - #"i) are amenable as well. Besides,Lemma 5.1 implies that

b&#+(1&&)#!,$A,"i'"!

i(#c + (1$ #)c", () * b#,$

A,"i(c, () · b#!,$

A,"!i(c", (),

with # = 12 . Hence

b12 #+

12 #!

(12c + 1

2c")* 1

2b#(c) + 12b#!(c"). (5)

This implies that the relation (4) holds for any dyadic rationnal #. The result forarbitrary # follows from the upper semi-continuity of b (Lemma 9.1).

10 Nontriviality of the entropy for products

Let F be a field. Let M be a closed F -orientable manifold, that is to sayHm(M ;F ) 2 F for m = dimM . In other words, either M is orientable, orF = Z2. Let fo : X = M! " R be a continuous function with range [0, 1].

Proposition 10.1 The associated homological entropy of fo achieves a strictlypositive value.

This results holds because a products M! inherits some Poincare duality (cf.Lemma 10.3) from the manifold M .

10.1 Poincare duality on a closed orientable manifold

Here follows the specific version of Poincare duality that is needed below.

Proposition 10.2 If a is a class in H"(M ;F ) whose restriction to the open setO does not vanish, then there exists a class b with support in O such that a,b 5= 0.

Proof. Let a ' Hi(M ;F ) with a|O 5= 0. Then there exists a homology class, ' Hi(O;F ) such that < a,, >5= 0. Let b be the Poincare dual of ,. Then a , bdoes not vanish since its evaluation on the fundamental class of M coincides with< a,, >. Moreover, if , is represented by a chain c, the class b can be representedby a form whose support is contained in any given neighborhood of the image of c.

10.2 Poincare duality in a product M!

Let id ' #o & ! be a finite subset and let A = A(#o) = p!"o(H"(M"o ;F )), where

p"o : M! " M"o is the canonical projection. Let also N = N(A) (cf. Assump-tion 3.1 and Example 3.3). If # & ! is another finite subset, we can define thepositive number

(" = ("(fo,#o) = 4 sup{55f"(x)$ f"(y)

55;x! = y! for ! ' # · #o}.

Observe that (" is decreasing in #. In fact (" approches 0 as # becomes large(cf. Lemma 10.4).

13

Lemma 10.3 (Poincare duality in M!) Let a belong to A". Then there exists alevel c"

a and an element b in A" such that

- supp a & f&1" ($%, c"

a + ("),

- supp b & f&1" (c"

a $ (",+%),

- a , b 5= 0.

Proof. The level c"a defined below obviously satisfies the first condition.

c"a = inf{c ' [0, 1]; supp a & f&1

" ($%, c)}. (6)

As explained below, existence of the class b follows from Poincare duality in anyfinite product M". Choose a point o in M! and define new functions g" : M! "[0, 1] by g"(x) = f"(x), with

x! =

6x! if ! ' # · #o

o! otherwise.

By definition of (",

supx%M"

55f"(x)$ g"(x)55 ( ("

4.

Now observe that the restriction of a to g&1" (c"

a $ 34(",+%) does not vanish.

Indeed, if it did vanish then supp a would be contained in g&1" ($%, c"

a $ 12(")

which itself is contained in f&1" ($%, c"

a $ 14("). This contradicts the definition

of c"a .

Since g" depends only on the variables indexed by #·#o, the open set g&1" (c"

a $34(",+%) coincides with the pullback p&1

"·"o(O) of some open subset O of M"·"o .

Combined with the fact that a = p!"·"oa for some class a in H"(M"·"o ;F ), this

implies that the restriction of the class a to O does not vanish. Poincare dualityin closed orientable manifolds yields a class b ' H"(M"·"o ;F ) with supp b & Oand such that a , b 5= 0. The class b = p!"·"o

(b) satisfies the required conditions.

The following result implies that in Lemma 10.3 one can replace (" by anygiven ( > 0 at the cost of considering only ”large” #’s.

Lemma 10.4 Let (#i)i(1 be an amenable sequence. Then the sequence ("i con-verges to 0.

Proof. Let ( > 0. By (uniform) continuity of fo, there exists a - = -(() > 0 suchthat d(x, y) < - 0 |fo(x)$ fo(y)| < (. The symbol d denotes one of the following(compatible) metrics on M! :

d(x, y) =!

!%!

do(x! , y!).|!| ,

14

where do is some Riemannian metric on M , where . is some fixed number in(1,+%) and where |!| = d(id, !). In particular d(x, y) < - when su$ciently manycomponents of x and y coincide. More precisely, there exists #% ' B(!) with#% 7 id such that d(x, y) < -(() as soon as x! = y! for all ! ' #%.

Now fix # = #i and let x, y ' M! be such that x! = y! for ! ' # · #o. Then55fo(!&1x)$ fo(!&1y)

55 < (

when ! ' intD(#·#o), where D denotes the diameter of #%. Set # = intD(#·#o).#and decompose f" into a convex linear combination as follows :

f" = |"||"|f" + |"&"|

|"| f"&".

Then55f"(x)$ f"(y)

55 (ˆ|"|

|"|55f"(x)$ f"(y)

55 + |"&"||"|

55f"&"(x)$ f"&"(y)55

( |"||"| ( + |"&"|

|"|( 2(,

provided the index i is su$ciently large. Indeed, #$ # & "D# = "D#i.

10.3 Repartition of classes according to degree and support

Now we are ready to prove the nontriviality of b. It follows from Lemma 10.3 andLemma 10.4 and does not further use the assumption that X is a product.

Proof of Proposition 10.1 Let A = A(#o) as before and let (#i) be an amenablesequence in !. Lemma 10.3 implies that for each i and each a ' A"i , there existsa c"i

a ' [0, 1] such that)"

A,"i,c!ia ,%!i

(a) 5= 0.

Thus any a in A"i contributes to b#,$A,"i,c,%!i

for some c and &. Using the pigeon-holeprinciple, in the spirit of Proposition 8.1, we will find some & and some c for whichexponentially many classes of degree around &|#i| are supported around f&1

"i(c).

The degree of a is an integer number between 0 and m|#o · #i| ( m/o|#i|,where m = dimM and /o = |#o|. Fix r ' No. Then for each i, there exists aninterval Ji = [ s&1

r , sr ] & [0, m/o] for which the rank of the space AJi

"iof classes in

A"i whose degree belongs to the interval Ji|#i| =, (s&1)

r |#i|, sr |#i|

-satisfies

rankAJi"i* 1

rm/orankA"i .

Lemma 10.5 Fix k * 1 and i * 1. Then there exist an interval Ii = [ j&1k , j

k ] &[0, 1] and a subspace Ai & AJi

"isuch that

- a ' Ai 0 c"ia ' Ii,

15

- rankAi *1k

rankAJi"i

.

Proof. If I & [0, 1] and if a is a cohomology class in H"(M!;F ), denote by rI(a)the restriction of a to the open set f&1

"i(I). Let [0, 1] = -k

j=1Ij , with Ij = [ j&1k , j

k ].We decompose the space AJi

"iinto a direct sum

AJi"i

= A1 6 . . .6Ak

in such a way as to satisfy the following properties :

Ak 6(Ker rIk .AJi

"i

)= AJi

"i,

and for j = 1, . . . , k $ 1,6

Aj &(Ker rIj+1'...'Ik .AJi

"i

)

Aj 6(Ker rIj'...'Ik .AJi

"i

)=

(Ker rIj+1'...'Ik .AJi

"i

).

Thus if a ' Aj then c"ia ' Ij . Now there exists a j = j(i) such that rankAj(i) *

1k rankAJi

"i. Let Ai = Aj(i).

The collection of intervals Ji and Ii being finite, there exist

- a subsequence of (#i), denoted (#i) as well,

- an interval J = [&$ 12r , & + 1

2r ], with & = &(r),

- an interval I = [c$ 12k , c + 1

2k ], with c = c(k),

such that ("i ( 14k , Ji = J and Ii = I for all i. Moreover, for each i and each

a ' Ai,supp a & f&1

"i($%, c"i

a + ("i) & f&1"i

($%, c + 1k ).

Furthermore, Lemma 10.3 provides a class b ' A"i such that

supp b & f&1"i

(c$ 1k ,+%) and a , b 5= 0.

Thus the map )#, 1

2r

A,"i,c, 1k

is injective on Ai for all i. Therefore,

b#, 1

2rA (c, 1

k ) * limi#$

1|#i|

ln7rankAi

8

* limi#$

1|#i|

ln(1

k

1rm/o

rankA"i

)

= limi#$

1|#i|

ln(1

k

1rm/o

7rankH"(M ;F )

8"i·"o)

= limi#$

|#i · #o||#i|

ln7rankH"(M ;F )

8

* ln7rankH"(M ;F )

8.

16

To conclude, observe that & = &(r) and c = c(k). Let ', ( > 0. There exists asubsequence &(rs) of &(r) (respectively c(kl) of c(k)) such that &(rs) (respectivelyc(kl)) converges to some & ' R+ (respectively c ' [0, 1]). Since b#,$

A,"(c, () increaseswith the size of (&$ ', & + ') and that of (c$ (, c + (),

b#,$A (c, () * b

#(rs), 12rs

A (c(kl), 1kl

) * ln7rankH"(M ;F )

8,

for l and s su$ciently large. Thus

b#(c) * ln7rankH"(M ;F )

8> 0.

Remark 10.6 At least for products, b#A(c) = b#(c) provided A contains H"(M ;F ).

Indeed, let Ao = p!!H"(M ;F ), some ! in !, and let A be another finite-dimensionalsubalgebra of A containing Ao, necessarily contained in some subalgebra A1 =p!"1

H"(M"1 ;F ) with #1 7 !. Thus Ao" & A" & A1

" and

b#,$Ao,"(c, () ( b#,$

A,"(c, () ( b#,$Ao,"(c, () +

7rankH"(M ;F )

8|"·"1&"|.

Thus

b#,$Ao (c, () ( b#,$

A (c, () ( max2

b#,$Ao

(c, () ,

limi#$

|"i·"1&"i||"i| ln

7rankH"(M ;F )

83= b#,$

Ao (c, ().

Example 10.7 Let Fo : S1 " [0, 1] be a Morse function with two non-degeneratecritical points, x&, the minimum, and x+, the maximum. Let fo = Fo 3 pid. Theentropy of fo can be computed explicitely (cf. [4], §A4). Indeed, let µ denote thefundamental class of M = S1. Then any class in H"(M";F ) is of the type

!

"!)"

n"!µ"!,

where n"! ' F and where µ"!= ,!%"!(p!!µ). Moreover, for a monomial a = µ"!

,9:

;c"a =

|#$ #"||#| ,

deg a = |#"|.

The first equality is a consequence of the fact that the fundamental class µ canbe supported in an arbitrarily small neighborood of any point (e.g. x&), implyingthat the class µ"!

can be supported in any neighborhood of M!&"! ! {x&}"!. It

is now easy to convince oneself that if A = p!idH"(M ;F ), then

b#,$A,"(c, () = #

<#" & #;

9==:

==;

|#$ #"||#| ' (c$ (, c + (),

|#"||#| ' (&$ ', & + ').

>.

17

Let n = |#|. Then

b#,$A,"(c, () =

!

(c&%)n<j<(c+%)n(1&#&$)n<j<(1&#+$)n

&n

j

'.

Therefore

sup(c&%)n<j<(c+%)n

(1&#&$)n<j<(1&#+$)n

&n

j

'( b#,$

A,"(c, () ( 2(n sup(c&%)n<j<(c+%)n

(1&#&$)n<j<(1&#+$)n

&n

j

'.

Besides, by Stirling’s formula,

1n

ln&

n

j

'8 1

nln

( nn+ 12

jj+ 12 (n$ j)n&j+ 1

2

)

8 $7 j

n

8ln

7 j

n

8$

71$ j

n

8ln

71$ j

n

8.

where we have removed terms that would produce a nul contribution in the limit.Now

b#,$A (c, () = sup

c&%<x<c+%1&#&$<x<1&#+$

($x lnx$ (1$ x) ln(1$ x)

).

And thus (using Remark 10.6)

b#(c) = b#A(c) =

?$% if c 5= 1$ &$c ln c$ (1$ c) ln(1$ c) if c = 1$ &.

So b is concentrated along the diagonal c = 1$ & and vanishes at the corners (0, 1)and (1, 0). The sum of the Betti number entropy is therefore given by

b(c) = $c ln c$ (1$ c) ln(1$ c).

Remark 10.8 As suggested by the previous example, it is always true in theproduct case that, provided the functions f" have constant range, the functionb(c) is nonnegative (i.e. does not achieve the value $%). This is a consequence ofthe presence of the fundamental class whose support can be concentrated aroundany given point in M .

11 Generalized Poincare duality

It has been observed in the product case that a class a in A", whose restrictionto an open subset O does not vanish, admits a nontrivial pairing with a class bin A" provided ”O is not too small”, meaning is of type p&1

" (Oo) for some Oo inM" (in the case A = p!idH

"(M ;F )). This suggests that a condition generalizingPoincare duality (more precisely its Lemma 10.3 version) should involve a filtration(T ")"%B(F ) of the topology T of X such that Poincare duality holds in A" foropen subsets of T " (a precise statement follows). In the product case the topology

18

T " is the one generated by the supports of the classes belonging to A". It seemsnecessary to assume in addition that these topologies are induced by a familyof (continuous) maps (X, T ") " (X, T ) converging uniformely to the identitymap. In the product case, for A = p!idH

"(M ;F ), these maps are the compositionsso" 3p", where so

" is a section M" " M! associated to a point o ' M! as follows :

7so"(x)

8!=

?x! if ! ' #o! if ! /' #.

A last detail : Lemma 10.3 holds when A is the full cohomology algebra of a finiteproduct. If A is not of this type (e.g. M = T2 and A is generated by one coho-mology class in H1(T2)), then the class b does belong to B" instead of A", whereB = H"(M ";F ) and # is the smallest set for which H"(M ";F ) / A.

Before stating the condition, we introduce the convention that whenever a se-quence something" (thus indexed by the set B(!)) is said to converge to something,it means that something"i converges to something whenever (#i) is an amenablesequence in !.

Condition 11.1 For any finite-dimensional subalgebra A & A, there exist an-other finite-dimensional subalgebra B with A & B & A and a family of continuousmaps (r" : X " X)"%B(!) such that if T " denote the topology obtained by pullingback that of X via r", then

- ! 3 r" = r!·" 3 !,

- r" converges uniformly and monotonously to the identity map,

- for any a ' A" and O ' T " such that a|O 5= 0, there exists a class b ' B"

with ?supp b & Oa , b 5= 0.

When, in addition to Assumption 3.1, the previous condition is fulfilled, one maycarry through the proofs of Lemma 10.3 and of Lemma 10.4, and hence that ofProposition 10.1.

Let A be a finite-dimensional subalgebra of A. Let ( > 0. Define g" = f" 3r" :(X, T ") " [0, 1] and let

(" = 4 sup{|f"(x)$ g"(x)|;x ' X}.

Lemma 11.2 (Poincare duality under Condition 11.1) If a belongs to A" forsome # ' B(!), then there exist a level c"

a and a class b in B" such that

- supp a & f&1" ($%, c"

a + ("),

- supp b & f&1" (c"

a $ (",+%),

- a , b 5= 0.

19

Proof. As in the proof of Lemma 10.3 let

c"a = inf{c ' [0, 1]; supp a & f&1

" ($%, c)}.

Now define O1 = f&1" (c"

a $ (",+%) and O2 = f&1" (c"

a $ 12(",+%). By construc-

tion,

supx%X

555f"(x)$ g"(x)555 =

("

4.

Hence

O2 &7g"

8&1(c$ 34(",+%) & O1.

Let O denote (g")&1(c$ 34(",+%). Then a|O 5= 0. Since O ' T ", there exists a

class b ' B" with supp b & O and a , b 5= 0.

Lemma 11.3 The sequence (" converges to 0. In other words, the sequence offunctions g" $ f" converges uniformly to 0.

Proof. Let ( > 0. Since r" converges uniformly and monotonously to the identitymap and since X is compact, there exists a finite set #o & ! containing id suchthat for any # / #o for which the amenability ratio #(#,#o) is su$ciently small,

555f(r"(x))$ f(x)555 < ( 9x ' X.

Let Do = diam#o. If ! ' intDo #, then555f(!&1r"(x))$ f(!&1x)

555 =555f(r!$1"(!&1x))$ f(!&1x)

555 < (.

Let "# = intDo #. Then555fe"(r"(x))$ fe"(x)

555 (1|"#|

!

!%e"

555f(!&1r"(x))$ f(!&1x)555 < (.

Hence

555g"(x)$ f"(x)555 ( |"#|

|#|

555fe"(r"(x))$ fe"(x)555

+|#$ "#||#|

555f"&e"(r"(x))$ f"&e"(x)555

( |"#||#|( +

|#$ "#||#|

( 2(,

Where the very last equality holds when, once more, # and #o have a su$cientlysmall amenability ratio.

Combining Lemma 11.2 and Lemma 11.3 we obtain the following result, whoseproof is essentially the same as that of Proposition 10.1.

Proposition 11.4 The function b achieves a strictly positive value.

20

12 Non-product example

Let M be a projective algebraic variety, e.g. the projective space CPn, and letY be a symbolic algebraic subvariety in X = M! (in the sense of [1] and [2]),that is, a compact subset Y such that Y", the ”restriction” of Y to each # inB(!), defined as the image of the natural projection M! " M", is an algebraicsubvariety in M". So Y comes as the projective limit of the Y"’s, where onemay (or may not) assume that Y is !-invariant. We assume that for large enoughd(#,#") (depending on Y ), the projection Y" - Y"! " Y"'"! is onto. Observethat surjective maps between projective (in general Kahler) varieties are injectiveon the top-dimensional cohomology with complex coe$cients due to existence ofmultivalued algebraic sections. (In fact if the fibers of such a map have, in asuitable sense, degree d, the same injectivity holds for Fp-coe$cients, provided pdoes not divide d). Therefore, if the target variety is non-singular, then the mapis injective on all cohomology by Poincare duality.

Subexample 12.1 Let M = CPn and consider a hypersurface in M !M repre-sented by an equation h(x, x") = 0. Then the infinite chain of equations h(xi, xi+1) =0, i = ....,$1, 0, 1, ... defines a subvariety Y in X = MZ invariant under the Z-action.

Remark 12.2 Unfortunately, even for generic h, it is unclear whether this Y isnon-singular in the sense that the restrictions of Y to the intervals [i, i+1, ..., i+k],denoted Y[i,i+1,...,i+k], are non-singular. However, a small (but non-Z-invariant)perturbation Y " of such a Y , allowing di"erent h’s, i.e. equations hi(xi, xi+1) = 0, isnon-singular by a simple argument (see [1] and [2]). Furthermore, the non-singularpertubations of Y are all canonically homeomorphic and thus their cohomologycan be attributed to Y (alternatively, one may speak of a random Y in X with asuitable Z-invariant probability measure on the space of strings {hi} and similarlyintroduce random potentials on Y (and/or on X itself). This significantly addsto possible examples and needs only a minor modification of our setting (with areference to the sub-additive ergodic theorem).

Continuation of the example. The cohomology of our (desingularized) Yenjoys the above product-like action on cohomology. In particular, for ! = Z, thehomological entropy of a function exists.

Remark 12.3 It seems hard to compute the (co)homologies of the above Y[i,i+1,...,i+k]

or even to elucidate the properties of (the analytic continuation of) their entropiclimit. However, it is easy to calculate the Chern numbers and thus the Eulercharacterictics of the Dolbeaut (and thus the ordinary) cohomology of all (desin-gularized!) Y[i,i+1,...,i+k].

13 Poincare polynomial

This section consists of defining the entropic Poincare polynomial of a !-space.It does not require the action to be product-like nor the group to be tileable.

21

Amenability is the only condition required here.

Consider the A"-Poincare polynomial of X :

pA,"(t) =$!

d=1

td rankAd",

where Ad" denote the set of classes in A" of (exact) degree d.

Lemma 13.1 The Poincare polynomial of X is !-invariant and subadditive, that is

pA,"'"!(t) ( pA,"(t) pA,"!(t) for t * 0. (7)

Proof. First observe that for any #1,#2 ' B(!), the map$

d1+d2=d

Ad1"1+Ad2

"2" Ad

"1'"2

is surjective. Thus

rankAd"1'"2

(!

d1+d2=d

rankAd1"1

rankAd2"2

,

which immediately implies the relation (7).

Thus the limitlim

i#$

1|#i|

ln7pA,"i(t)

8

exists whenever (#i) is an amenable sequence in ! (cf. [3]). We define the Poincarepolynomial of the group action $ : !!M " M to be

p(t) = supA

limi#$

1|#i|

ln7pA,"i(t)

8.

Remark 13.2 This definition is analoguous to that in [1] §1.14. Indeed, theprocess of factoring away *-fillable classes corresponds roughly to restricting toclasses in A".

References

[1] Misha Gromov, Topological invariants of dynamical systems and spaces ofholomorphic maps. I. Math. Phys. Anal. Geom. 2 (1999), no.4, 323–415.

[2] Misha Gromov, Endomorphisms of symbolic algebraic varieties. J. Eur. Math.Soc. 1 (1999), no. 2, 109–197.

[3] Donald S. Ornstein, Benjamin Weiss, Entropy and isomorphism theorems foractions of amenable groups. J. Analyse Math. 48 (1987), 1–141.

22

[4] Oscar Lanford, Entropy and Equilibrium States in Classical Statistical Me-chanics. Statistical mechanics and mathematical problems. Battelle Rencon-tres, Seattle, Wash., 1971. Lecture Notes in Physics, 20. Springer-Verlag,Berlin-New York, 1973.

[5] Weiss, Benjamin, Monotileable amenable groups. Topology, ergodic theory,real algebraic geometry, 257–262, Amer. Math. Soc. Transl. Ser. 2, 202, AMS,Providence, RI, 2001.

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