< • · -PDc:5D'D5 Coupled Analytic Maps J.Bricmont* UeL, Physique Theorique, B-1348, Louvain-la-Neuve, Belgium A.Kupiainen t Helsinki University, Department of Mathematics, Helsinki 00014, Finland Abstract We consider a lattice of weakly coupled expanding circle maps. We construct, via a cluster expansion of the Perron-Frobenius operator, an invariant measure for these infinite dimensional dynamical systems which exhibits space-time-chaos. 1 Introduction It is an important problem to determine which parts of the rich theory of finite di- mensional dynamical systems (e.g. hyperbolic attractors, SRB measures [11]) can be extended to infinite dimensional ones. The latter are usually given by non-linear partial differential equations of the form 8,u = F(u, 8u, 8 2 u, ... ), i.e. the time derivative of u(x, t) is given in terms of u(x, t) and its partial space derivatives. One would like to find natural invariant measures for the flow. In a bounded spatial domain and F suitably dissipative, such equations tend to have finite dimensional attracting sets [26J and thus fall in into the class of finite dimensional systems. Genuinely infinite dimensional phenomena are expected to occur for dissipative PDE's on unbounded domains [9J. In particular, invariant measures for the flow might pave infinite dimensional supports and there might be several of them (corresponding to a "phase transition"). A class of dynamical systems, possibly modelling such PDE's, are obtained by dis- cretizing space and time and considering a recursion u(x, t + 1) = F(x, u(-, t)) (1) i.e. u(x, t + 1), with x being a site of a lattice, is determined by the values taken by u at time t (usually on the sites in a neighbourhood of x). For a suitable class of F's such dynamical systems are called Coupled Map Lattices [14, 15]. 'Supported by 8C grants SCI-CT91-0695 and ISupported by NSF grant DMS-9205296 1
Helsinki University, Department of Mathematics,Helsinki 00014, Finland
Abstract
We consider a lattice of weakly coupled expanding circle maps. We construct,via a cluster expansion of the Perron-Frobenius operator, an invariant measure forthese infinite dimensional dynamical systems which exhibits space-time-chaos.
1 Introduction
It is an important problem to determine which parts of the rich theory of finite dimensional dynamical systems (e.g. hyperbolic attractors, SRB measures [11]) can beextended to infinite dimensional ones. The latter are usually given by non-linear partialdifferential equations of the form 8,u = F(u, 8u, 82u, ...), i.e. the time derivative ofu(x, t) is given in terms of u(x, t) and its partial space derivatives. One would like tofind natural invariant measures for the flow.
In a bounded spatial domain and F suitably dissipative, such equations tend to havefinite dimensional attracting sets [26J and thus fall in into the class of finite dimensionalsystems. Genuinely infinite dimensional phenomena are expected to occur for dissipativePDE's on unbounded domains [9J. In particular, invariant measures for the flow mightpave infinite dimensional supports and there might be several of them (correspondingto a "phase transition").
A class of dynamical systems, possibly modelling such PDE's, are obtained by discretizing space and time and considering a recursion
u(x, t + 1) = F(x, u(-, t)) (1)
i.e. u(x, t + 1), with x being a site of a lattice, is determined by the values taken by uat time t (usually on the sites in a neighbourhood of x). For a suitable class of F's suchdynamical systems are called Coupled Map Lattices [14, 15].
'Supported by 8C grants SCI-CT91-0695 and Cf:l~"(-CT93-0411
ISupported by NSF grant DMS-9205296
1
The first rigorous results on such systems are due to Bunimovich and Sinai whostudied a one dimensional lattice of weakly coupled maps [5]. They established theexistence of an invariant measure with exponential decay of correlations in space-time.Their method was to construct a Markov partition and to show uniqueness of the Gibbsstate for the corresponding two-dimensional spin system. This is a natural extension ofthe method used for a single map or for hyperbolic systems [25, 231. These results werestrengthened by Volevich [27] and extended by Pesin and Sinai to coupled hyperbolicattractors [20] (for a review, see [6]). An extension to lattices of any dimension isannounced in [28] (for coupled hyperbolic attractors).
Since the Gibbs measure constructed by Bunimovich and Sinai describes statisticalmechanics in two dimensions, the possibility of phase transitions i.e. non-uniqueness ofinvariant measure is open (for recent results on this, see [1, 2, 4, 7, 19, 22] and referencestherein). In statistical mechanics, Gibbs measures are often easy to construct in weakcoupling (which corresponds to high temperature) and in strong coupling (low temperature) using convergent expansions. The purpose of the present paper is to develop theseexpansion methods for the dynamical system problems in infinite dimensions.
We consider weakly coupled circle maps and derive a convergent cluster expansion forthe Perron-Frobenius operator (transfer matrix in the statistical mechanics terminology).This allows us to prove exponential mixing in space and time for an invariant SRBmeasure. These results are similar to those of Bunimovich and Sinai, but our methodworks immediately in any dimension and is simpler. However, for technical reasons weneed to restrict ourselves to real analytic maps.
The Perron-Frobenius operator has been a powerful tool to analyze quite generalmaps, of bounded variation [18] (for reviews see [8, 17]). This approach was used alsofor coupled maps in [16], but weaker results were obtained there in the infinite volumelimit. An open problem still remains to develop expansion methods for coupled mapsthat are of bounded variation. These are the most natural candidates that might exhibitinteresting phase transitions as the coupling is increased.
We have tried to make the paper self-contained for readers having no backgroundin the expansion methods of statistical mechanics. Appendix 2 contains some of thestandard combinatorical estimates needed. A reader who is familiar with these methodswill find a slightly novel application of them because our expansion is applied directlyto the Perron-Frobeni us operator.
2 Results
We consider the following infinite dimensional dynamical system. The state space of thesystem is
M = (81)Zd,
the direct product of circles over Zd, i.e., m EM is given as m = {m;}ieZd, mi E 81.M carries the product topology and the Borel a-algebra inherited from 8 1. To describethe dynamics, we consider a map f : 8' ---t 8' and let :F : M ---t M be:F = xieZd f i.e.,
2
·.
F(m); = f(m;). F is the uncoupled map. The coupling map <Ii : M ---+ M is given by
(2)
where gn : 51 X 51 ---+ R (g will be chosen such that the sum converges) and E > 0 is aparameter. We define now the coupled map T : M ---+ M by
T=<lio:F. (3)
We assume the following:
A. f is expanding and real analytic.
B. gn are exponentially decreasing and real analytic.
More precisely, for B we assume that there exists a neighbourhood V of 51 in C suchthat gn are analytic in V x V and
(4)
for some C < 00, A > 0 and all u, v Eli.
Let us denote by 0"; the shifts in Zd, i = 1, ... , d. Then our main results are
Theorem. Suppose f and gn satisfy A, B. There is an Eo > 0 such that for E < Eo
there exists a B01'el meaSU1'e It on }\II such that
1. p, is invariant unde1' T and the shifts 0";. F01' any finite A C Zd, the ma1'ginaldistribution of p, on (51)A is absolutely continuous with respect to the Lebesgue meaSU1·e.
2. The Zd+1 action generated by T and {0";}1=1 is exponentially mixing.
3. Tnm ---+ p, weakly as n ---+ 00, where m is the product of Lebesgue meaSU1'es on 51.
Remark 1. For a precise statement of 2, see Proposition 7. Properties 1 and 2 areusually called "space-time chaos".
Remark 2. The result holds for a much more general class of interactions <Ii : we couldtake
<Ii(m); = m;e2~;Lx'iexgX(mx)
where X C Zd, mx = mix, IXI is finite and gx : (51)IX 1 ---+ R is analytic in Vixi withthe bound
Igx(zx)1 ~ E1X1e-h(X)
where r(X) is the length of the shortest tree graph on the set X. Then the theorem holdsagain for E < Eo small enough provided gx +j = gx, j E Zd Without this translationinvariance It is not O";-invariant, but the other claims still hold. Similarily, F may bereplaced by x ;Ezdf; where f; satisfy A, uniformly in i. Our results also extend to coupledmaps of the interval [0,1], of the type considered by Bunimovich and Sinai [5], providedwe take their f and Q analytic.
Remark 3. We do not prove, but conjecture, that there is only one invariant measurewhose local marginal distributions on the sets (51)A, A C Zd,IAI finite are absolutely
3
'-.-'
continuous with respect to Lebesgue measure. This is true in a class of measures withanalytic marginals: the proof of 3 extends to the case where m is replaced with a measuresatisfying some clustering and the analyticity of the densities of the local marginaldistributions. This is similar to the results of Volevich [27,28]. Thus JL can be consideredas a natural extension to the infinite dimensional context of an SRB measure.
3 Decoupling of the Perron-Frobenius operator
Let A C Zd be a finite connected set (a set A in Zd is connected if every point of A has anearest neighbour in A, or if A consists of a single point). We denote M A = (51)A Letnow IA = XiEAI and let <I>A : Jlth --t J\lh be given by (2) where the sum is over j in A.We first construct a TA = <I> A 0 IA invariant measure JLA on J\lh and later JL as a suitablelimit of JLA. The measures JLA will be constructed by means of the Perron-Frobeniusoperator of TA • To describe this, wel1rst collect some straightforward facts about I andits invariant measure.
The Perron-Frobenius operator P for I on L1(5') is defined as usual by
J(g 0 J)h dm = JgPh dm
for hE L 1(51),g E Loo(51 ) and dm the Lebesgue measure (i.e., dx in the parametrizationm = e2..
iX). We wish to consider P on a smaller space, namely Hp , the space of bounded
holomorphic functions on the annulus A p = {1- P < Izi < 1+ p}. Hp is a Banach spacein the sup norm. The assumption A for I implies the following spectral properties forP:
Proposition 1. There are constants Po > 0, I' > 1 such that
a) P : H p --t H-,p is continuous fOl' all p :::; Po.
b) P : Hp --t Hp can be written as
P=Q+R
with Q a i-dimensional projection opemto''';
Qg= (Lgdm)h=e(g)h
with h E Hpo,h > 0 on 5', lSI hdm = 1 and
III R n III:::; CJLn, QR = RQ = 0
(5)
(6)
(7)
for some JL < 1, C < 00 and all n; we use III . III to denote opemtol' norms.
Remark 1. a) is a consequence of the expansiveness of I and shows that the operatorP i~proves. the domain of analyticity, while b) means that there is a unique absolutelycontlOuous IOvarlant measure for I, with density in H po ' and the rest of the spectrum of
· ~
Pin ffpo is strictly inside the unit disc. Since all this is rather standard (see [8, 17, 18]),we defer the proof to Appendix 1.
Remark 2. Throughout the paper C will denote a generic constant, which may changefrom place to place, even in the same equation.
To describe the Perron-Frobenius operator PA for the coupled map TA , we introducesome notation. We denote 'H.~ = 0 iEA H p , P A = 0 iEA P i where Pi acts on the i-thvariable. Also, we denote by dmA the Lebesgue measure on M A. Then
where
and thus
(ili~G)(m) = detDili;;-'(m)G(ili;;-l(m))
(8)
(9)
(10)j(Go1"A)H dmA = j GPAff dmA
for G E LOO(MA),H E LI(J\lh). We have o[ course to show that (9) is well defined.Actually, the strategy of our proof will be to first derive a "cluster expansion" [12, 13Jfor iliA in terms of localized operators with good bounds on norms. Then we shallconstruct the invariant measure in Section 4 by studying the limit n --> 00 of Ph; acluster expansion for Ph will be obtained by combining (5) for PA and (11) below for
iliA'
Proposition 2. Let iliA given by (2)with i,j E A, and let gn satisJy B. The Jollowingholds uniJormly in A : The,'e exists PI > 0 and co > 0 such that Jor c < co, iliA maps'H.~, into 'H.~,_s where we may take 0 --> 0 as c --> O. !VIo,'eover
ili~ = L 0yEYOY 0 1A\yy
(11)
whe,'e Y runs though sets oj disjoint subsets oj A, A \ Y = A \ uY and 1z denotes theimbedding oj 'H.:, into 'H.:,_s. The opemtors 01' : 'H.~ --> 'H.~,_s are bounded, with
(12)
when the sum nms over tree graphs on Y, ITI is the length oj T and '7 --> 0 as c --> O.
Proof. Let us introduce decoupling parameters s = {Sij}, i,j E A, i < j in some linearorder of A, [or our map ili~:
(13)
where
(14)
5
. '._.
and we let Sii =1, Sij = Sji. Our assumption B for g" implies that there exists a PI > 0such that all g,,'s are holomorphic in ApI x API ( Ap is the annulus) and (4) holds there,together with
(15)
where a = a/au or a/av. Consider now S complex, in the polydisc D/I. = {sijli <j, ISijl < Te~li-jl} C ct<l/l.I'-I/l.1) (we shall take I' large for c; small). Then, we have thefollowing
Lemma. Thel'e exists c;o(Pl, I', A) sitch that, JOT c; < C;o(p" I', A) <I>;;:~ is a holomoTphicJamily (JOI'S E D/I.) oj holomoTphic diffeomoTphisms JTOm A~,_s into A~, wheTe 0 -+ 0as c; -+ 0 (and A~ denotes the polyannltllts {Zi E A p , i E A}).NIol'eoveT, the bound
II det D<I>;;:~ II ::; exp(Cc;I'(l + A-d) IA I)
holds uniJmmly in A. II· II is the nm'm in 'H~,_S'
Proof. (4) and (15) imply
(16)
(17)
(18)
for Z E A~, and S E D/I.. From these inequalities it follows easily that <I>/I.,. is a diffeomorphism from A~, onto a set containing A~,-S provided C(l + A-d)C;T < O. The inverse of<I>/I.,. then satisfies bounds (17), (18) too (with different constants) and then (16) followsfrom Hadamard's inequality [10J. 0
Thus, <I>A. : 'H~, -+ 'H~,-S is a halomorphic family of operators, for sED/I..
The cluster expansion for <I>~ is the following repeated application of the fundamentaltheorem of calculus:
(19)
where the notation is as follows. r is a subset of Ax A and each pair (i,j) E r issuch that i < j. ds r =IT(i,j)Er ds ij , ar = IT(i,j)Er a~i)' Sr< = {Sij} (i,j)Er< and Irl is the
cardinality of r. (19) then follows by writing 1 = ITi<j(Iij + Eij ) with Iij = f[olJ dSija~i)
and E ij is the evaluation map at Sij = 0 (for more details, see [12] and Proposition 18.2.2in [13]).
Now observe that OAr factorizes as follows. For a set A C A x A, let A denote theunion of its projections on the two factors. We say PI, pz E r are connected to eachother if 15, n15z =I 0. Let rex be the maximal connected components of r with respect tothis relation. Then we have
(20)
6
· "
where "['ofo : Hr; -t Hr~-s is given by the integral in (19) (with Areplaced by fa andfbyfa ).
We now estimate the norm of "['ofo ' Using (16) (which holds for arbitrary A) and a
Cauchy estimate in (19) (a circle of radius ~e~I;-jl around any S;j in the integral (19) isinside D A ), we get
(21)
We arrive at the final formula (11) by setting
(22)
with Y = f and f connected. Given any such f which we may view as a connectedgraph on the points of Y, pick a connected tree graph T with T = Y and estimate thesum over f in (22) by
III"FIlI ::; (L e-t'\ljl)IYI~1Y1-I L e-~ITIjEZd T
where ~ = r- 1 exp[Cc:r(l + A-d)] (see Appendix 2 for more details).,. -t 00 as c: -t 0, see (16-18), the claim (12) follows.
(23)
Since we may takeD
Remark. Propositions 1 and 2 imply that there is an C:o > 0 and p > 0 such that PA
maps H~ into itself for all A and all c: ::; C:o: the domain of analyticity shrinks by anamount" when we apply <Pi., but it is expanded when we apply PA (by Proposition 1,a).So, we may choose c: small enough so that,p - " > p. We will fix this p now once andfor all.
4 Space time expansion for the invariant measure
The TA-invariant measure ItA will be constructed by studying P:\ as n -t 00. This willyield spectral information on PA uniformly in A. From (8), we have
(24)
into which we insert (11) for <Pi. and, for PA, we use (5):
PA = (/9iEA (Qi + R;) = L (/9;EI R; (/9jEA\I Qj == L R[ (/9 QA\I' (25)leA leA
and we get
n
P:\ = L L II((/9FEY, "F (/91 A\y,)(R1, (/9 QA\I,){I,) (Y,) '=1
7
(26)
· ,
where the product of operators is ordered, with t = 1 on the right.
To understand the structure of the terms in (26), let us first consider a simple example.Consider the term in (26) with Yt = 0 for t # m < nand Ym = {Y}, where Y is somefinite subset of Zd Let also It = {i} for 1 ::; t ::; m and It = {j} for m + 1 ::; t ::; n,where i,j E Y. The contribution to (26) from this term is given by
where we used Q2 = Q repeatedly. Let us write Q in (6) as
Q= hr2;£
(27)
(28)
where we identify bounded operators Iip ---+ Hp with elements of Hp tensored with itsdual H; and use' to distinguish this tensoring from the one used for spaces indexed byZd. Using the shorthand
£1' == I8!;El' £, Iw = l8!iEY h
and inserting (28) into (27), the latter becomes
HI 18! (hA\jr2;£A\i)
where HI : '}-{~;} ---+ '}-{~j} is given by (let 9 E '}-{~;})
Hlg =£y\j (Rj-moy(hY\i 18! R'('g)).
To make these remarks systematic, it is useful to introduce a "space-time" lattice Zd+l,where the extra dimension corresponds to the "time" t in the product in (26), and toestablish a correspondence between the terms in (26) and geometrical objects definedon this lattice.
We let S denote the set of all finite "spacelike" subsets of Zd+l, i.e. Z E S is of theform Z = Y x {t} for some Y C Zd and t E Z. Also, let B denote the set of "timelikebonds" of Zd+l, i.e. & E B is of the form &= {(i, t), (i, t + 1)} == &;(t) for some i E Zdand integer t. The correspondence with terms in (26) is defined as follows: to eachY E Yt, t = 1,···, n, we associate Z = Y x {t} and to each It we associate the set ofbonds {&;(t - 1), i E Id.A polymer, is then defined as a connected finite subset of SUB, i.e. the elements of, are spacelike subsets and timelike bonds of Zd+l. We define two bonds &, &' E , tobe connected, if &n &' # 0 01' if there exists a Z E , such that & and &' intersect Z., is then defined to be connected if the set of bonds & E , is connected with respectto this relation, or if , consists of a single element belonging to S. Denote by "I thesupport of " i.e. the subset of Zd+1 that is the union of the elements of ,. We saythat two polymers are disjoint, if "II n "12 = 0. Thus, in the example above we have,= {Y x {m},&;(O)'''',&i(m -l),&j(m), ... ,&j(n -1)}.
Let, be a polymer. Denote by 71', and 71', the projections on Zd+l = Zd X Z to the firstand the second factor. Then 71't('f) is connected, i.e. it is an interval denoted [L, t+]. Let
8
. .
, .
'Y± = 7rs (;yn 7r,l(t±)) The weight of the polymer "I, W("(), is a bounded linear operatorW("() : H;- -> H;+ or equivalently W("() E H;+0(H;-)*. W in the example above is aweight with "1+ = j and "1- = i.
We may now return to (26). Let us in general denote by Atlt, the set Atlt, = A X {t" t, +1, ... , t 2 } c Zd+1 where -00 :5 t, < t 2 :5 00 and A x {t} == At. "I is said to be in At" , if;Y c A'I'" and each Z E'Y has t l +1 :5 7rt(Z) :5 t2 • In (26) we will encounter a family ofpolymers f in AOn and we need to make a distinction whether their support intersectsthe boundary of Aon i.e. Ao or An. We denote the family of those intersecting neither byf v (v stands for "vacuum polymers"), those intersecting Ao but not An by f o, the onesintersecting An but not Ao by f n and the ones intersecting both by fOn (the example in(27) belongs to fOn)' Finally, let f1"( be the set of i E ;Y such that i belongs to exactlyone b E "I and to no Z E "I.
The convergence of the polymer expansion is due to two reasons: thanks to (12),each Oy brings a small factor, which decays with the size of Y or the distance betweenthe points of Y. On the other hand, bonds are associated to R factors and long stringsof such factors are suppressed by (7). Note however that J.L need not be small, only lessthan one. Thus, this expansion is similar to the one of a lattice of weakly coupled onedimensional systems, but where, within each system, the couplings are not necessarilysmall.
We have now
Proposition 3. (26)can be written as
P A= 'L II (W('Y)) 0.,eron W("()r -yEr"
o ((h A+ 0.,ern W("()h)0(eL 0.,ero ew("())) (29)
where the sum is ove,' sets f (possibly empty) oj mutually disjoint polymers "I withfJ"( C AoUAn. The Jollowing notation was used: ew("() = e.,+ W("() E (H;- t ,W("()h =W(,,()h.,_ E H;+ , (W("()) = e.,+ W("()h.,_ E Rand A+ = A\ U.,ernuron "1+ and A_ =A\ U.,erouron "1-.The weights satisJy
III W("() 111:5 J.LB II (C7))IZIT(Z,A)Ze.,
(30)
(31 )
where B is the number oj bonds in "I and III . III is the norm in H;+ 0(H;- t .Proof. Consider a given term in (26). {I,}?=, determines a set of bonds 8 0 = {bi(t) liE1,+, , t E [O,n -In and {Y,};'::l a subset of S, So = {Y x {t} , Y E Y,}. Decompose8 0 U So = U.,er'Y where f is a set of mutually disjoint polymers. Since QR = RQ = 0,we see moreover that 8"1 C Ao U An. Thus the sum in (26) can be written as
PA= 'L0r(A)r
where f runs through such sets and Or(A) is the product in (26) corresponding to f.Note also that we may, since Q2 = Q, replace each Q in Or(A) which is on a bond that
9
.' .
is disjoint from r and Ao U An by the identity operator. For the remaining Q's use theidentity
QKQ = e(Kh)Q, (32)
where K : tip ---> 7ip , to factorize Or(A): We apply (32) (extended to tensor products)to each product QibYQi in (26) where i E Y. The result is the summand in (29) wherethe weights W(-y) : 7i;- ---> 7i;+ are given by (here F E 7i;-)
(33)
Yin (33) is 1rs ('7) and D.y(Y) is given by the product in (26) with A replaced with Y.
To bound W(-y), we use (12) and (7), the bounds lIeli :::; 1, IIhll :::; C (the norm of linearfunctionals is also denoted by II . II) and the fact that on bonds b of Y x 1r,('7) that aredisjoint from '7, we have identity operators. 0
Equation (29) is an example of polymer expansion in statistical mechanics and it is wellknown (see e.g. [3, 21, 24]) that the bounds (30) will enable us to prove exponentialfalloff of correlations (i.e. mixing) and construct the A ---> Zd limit (in the standardtreatments the weights of polymers are scalars and not operators as here, but, as we willsee, the combinatorical part of the proof is as in the standard case). We refer the readernot familiar with the combinatorical methods needed in this analysis to the referencescited above and to Appendix 2 and just spell out the main steps here.
First we wish to cancel in (29) the contribution from the polymers 'Y E f v ' Let uscall these the vacuum polymers. These are "freely floating" in AOn- 1 unlike the othersthat are attached to the boundary Ao U An, but they tend to cancel each other. Tosee this, note that, by (10) with G = 1, eA pk = eA for all k, so, since e(h) = 1, andRh = RQh = 0 (which implies W(-y)h = 0 for 'Y E f o U fOn), we get
1 = eA(p~-lhA) = L: mW(-y))r .,
(34)
where f is a set of disjoint vacuum polymers in AOn - 1 (note that the vacuum polymersin (29) lie in AOn- 1 too). The cancellation we are after is accomplished by using (34) towrite
p~ = (L: mW(-y)))-lp~ (35)r .,
and substituting (29). The standard combinatorics (see Appendix 2) now yields
when the sum is over sets f (possibly empty) oj disjoint polymers 'Y in Aon with '7 n(Ao U An) # 0 and 8'Y C Ao U An. The weights satisJy
III V(-y) III:::; JiB II (C,/)lzIT(Z,A/2). (37)ZE.,
10
Notice now that, as n --t 00, the I E f On in (36) give exponentially small contributionsdue to (37), and (36) will factorize. Let us define a function DA E HZ
DA= :L>'Alr+ I8i..,Er IIb)hI'
(38)
where f is a set of finite, mutually disjoint polymers I in A- ooo with "I n Ao oF 0 andEh CAo. Similarily, define a linear functional £A : HZ --t R by
£A = L eAlr_18i"'Er ellb)r
where I are now in Aooo , with 0 oF Elf c Ao. Put
(39)
(40)
Then we have
Proposition 5. (Spectml decomposition oj PA). There exist co > 0, 1-'-1 < 1, c < 00,
independent oj A, such that, JOI' c < co, nAand LA have the Jollowing properties
IlnAII ::; eclA1 , LA = eA
PAnA= nAIII P~ - nA~LA III ::; I'~ eclAI
(41 )
(42)(43)
J01' all A C Zd and n EN.
Remark. Thus the spectrum of PA consists of the eigenvalue 1 with multiplicity 1 andthe rest is in the disc {izi ::; I'd, uniformly in A.
Proof. We have
and (using Ilhll ::; C)
eA(DA) = L II (lib))I' ..,
II DAII::; C IAI L II II lib) II .I' ..,
(44)
Since II ell::; 1, l(IIb)) I satisfies (37). Now standard estimates (see Appendix 2) give
(45)
which yields the estimate in (41).Consider next P~ - DA~ £A . Using (36),(38) and (39), this is given again by
(36), but with different constraints for the set f: either there exists a I such that'7 n Ao oF 0, '7 n An oF 0 or these exists a pair {t'I'} such that "I n "I' oF 0 and "I U "I'intersects both Ao and An. The first set of terms come from P:\ and are not canceledby the corresponding terms in nA~LA and the second set are the uncanceled terms
11
coming from (38,39). In both cases there is an overall p n factor and using (37) andagain standard estimates for the combinatorics (see Appendix 2) we get
(46)
Since'l -> 0 as c -> 0, (43) follows with PI = p(l + C'l). To show LA = fA recall that,from (10), fA P'A = fA for all n. This together with (43) gives
To prove (42), use (43) to get
as n -> 00, and then use the continuity of PA . oWe will pass shortly to the 1\ = Zd limit but before that we need a more refined mixingcondition than (43):
Proposition 6. Thel'e exists PI < 1, c < 00 such that, if dpA = fJAdmA denotesthe TA invariant meaSUl'e constructed in Proposition 5, then, for any X, Y c 1\ and
x yF E 1ip ,G E 1ip ,
Proof. We want to compare
(48)
withfA(FOA)fA(GO A).
To do this, we expand them. Let us denote the terms in the sum (38) by nA(f) and theones in (39) by lA(r). Then we get, see (40,'11),
fA(FO A) = lA(FnA) = I:lA(r)FnA(f')ff'
(49)
where f is a set in 1\000' f' in 1\-000 and their members satisfy')" n 1\0 # 0 and 0 # (Jy C
1\0,We want next to group together all ",' that "connect" to X. For this, consider the projections 1l".(')') == Y'Y of')' to 1\ and define Y'Y' similarily. Define any two sets in {X, y" Y'Y' }to be connected if they intersect. Let Y be the connected component including X underthis relation and f y , fj, the sets of, and " contributing to Y. We may then rewrite(49) as
fA(FfJ A ) = I: fA\y(OA\l') I: ly(fy )Fny(f~)Y:XcY fy,r;..
12
(50)
Since £z(D.z ) = 1 for all Z, (50) equals
£A(FD. A) = L lIF (r)['
(51 )
where f is a set of polymers I in A-ooo or in Aooo , intersecting Ao and with projectionson A connected to X in the above sense, and 1Ip(r) is the corresponding term in (51).£A(GD.A) has a similar expansion where the f's are connected to Y, so consider (48).We proceed as above, using (36):
eA(FP~(GD.A)) = L VF (fdlla (f2) +L lIF,a(f)rt nr2=0 r
(52)
where f = U~El' "1, the first sum comes from the f On = 0 terms in (36) and, in the secondone, we sum over sets f wi th the following properties: f = f -000 ufo u f n U f On U f noo,with by now an obvious notation, and fOn i 0; moreover, the projections 7fs b) are nowconnected to X and Y. lIF ,G is given by
The bound (47) follows now from (51) and (53): the left hand side of (47) has anexpansion like (52) but with f 1 n f 2 i 0 in the first sum and then each term in bothsums contributes fln e-+d(X,V) (as in the proof of (43)). The combinatorics is controlledby e,min(IXI,1Y1l because each f must be connected to X and Y (see Appendix 2). D
5 The A -------t Zd limit: Proof of the Theorem
We have now all the ingredients for the proof of the Theorem. First we describe theinvariant measure fl. To do this, rewrite nA in (38) as
fiA = L @~ Vb)h A['
with Vb) = V(~)h, . Since h > 0 on Sl (Proposition 1), h i 0 on Sp for p small enough'+
and thus liVll satisfies (37) too. Thus we may exponentiate (see Appendix 2)
fiA = exp(LUb))hA ,..,
U satisfies
Define, for YeA, Hy E 'H.~ by
Ifl' = L Ub),)'+=y
13
(54)
(55)
< •
where'Y C Z~ooo and let H},A be given by the same sum with 'Y C A- ooo . Then
(56)
and
(57)
where d denotes the distance. We have
It is easy to see that there is a unique Gibbs measure corresponding to the HamiltonianH, for TJ small. We take f.L to be this measure, i.e., the unique Borel probability measureon M with conditional probability densities given by
(d I ) exp[- Ll'nA;f0 H}'(m)] I df.L mA mAO = 'A mAJ exp[ - L}'nA;f0 HI' (m)]
(58)
for any A C Zd finite [23, 24].
The O"i-invariance follows since the unique Gibbs measure satisfying (58) is translationinvariant. For the T-invariance of f.L it suffices to show
J F 0 T df.L = J F df.L
for all F E H;, all X finite. Let T(A) = TA 181 fA' and df.L(A) = df.LA 181 hA,dmA'; then,f.L(A) is T(A) invariant. Moreover F 0 T is continuous and F 0 T(A) -> F 0 T in the supnorm. (56) and (57) imply that I,(A) -> f.L weakly (here A -> Zd is taken in the sense ofthe net of finite subsets of Zd, see [23, 24]). Hence,
J F 0 T dl' = lim J F 0 T(A)df.L(A) = J F df.L.
Mixing follows from (47) which carries over to the limit;
Proposition 7. Let T denote the Zd+l action generated by T and the O"i. There exista> 0, C < 00, such that for all finite X C Zd, all F, G E H;, and all n E Zd+l
Finally, to prove part 3 of the Theorem, it is enough, by a density argument, to show,
for allfini te X C Zd, and all F E H~Y that
lim J F 0 Tndm = JFdf.Ln_oo
14
· .
But,
and
So, it is enough to show
with Ill, c, independent of A. But this follows from (47) with G = hI!'
Acknowledgments
D
We would like to thank L.A. Bunimovich, E. Jarvenpaa, J. Losson and Y.G. Sinaifor interesting discussions. This work was supported by NSF grant DMS-9205296 andby EC grants SC1-CT91-0695 and CHRX-CT93-0411.
Appendix 1: Proof of Proposition 1
We work in terms of the lift of I to the covering space R of 5', and denote it by I again.Thus, our assumptions are: There exists a Po > 0, 1 > 1 such that
(a) I is holomorphic and bounded on 5po = {z E C1IIrnzl < Po} with I(R) = R,
(b) I(z + 1) = I(z) + k z E 5po ' kEN, k ~ 2,
(c)J'(X)~, xER.
Hence, by changing Po, 1 if necessary, we may assume 1(5p) :::l 5'"!p p::::; po and,p == I-I is holomorphic and bounded on 5'"!po and 1J'(z)1 ~ " 1,p'(z)1 ::::; 111 on 5po and5'"!po respectively. Let Iip be the space of periodic bounded holomorphic functions on5p , of period one (which can be identified with Iip of Sect.2). Since P is given by
(Fc )(z) = I: g (1* + j))g j=O 1'(,p(z+J))
(59)
we get from the above remarks that P : Iip -> Ii'"!p for p ::::; Po, which is the claim a) ofProposition 1.
15
· ,
The proof of the spectral decomposition b) goes via finite rank approximations. Let,pq = j-q and let Pqn : 11p --t 11p be the finite rank operator
By Taylor's theorem and Cauchy's estimates, we have, for z ESp, Rez E [0,1],
< ,-,q L l,pq(z + j) -,pq(jWlIg(n)llp/2n. j
< ,-q(n+l)kqcnp-nllgllp
where we assume ,pq(Sp) C Sp/2, which holds for q large enough and we used 11,p~lIp ~ ,-qin the second inequality. Since 9 is periodic, we may restrict ourselves to Rez E [0,1].
Take now first q large enough such that C p-l,-q ~ ~ and then n large enough suchthat kq2-n < ~. Then
III pq - Pqn 111< ~.
From this we conclude, since Pq" is of finite rank, that the spectrum of pq outside ofthe disc of radius ~ consists of a finite number of eigenvalues with finite multiplicities.Therefore the same holds for P outside of a disc strictly inside the unit disc. On theother hand it is well known that in a space of ['-functions of bounded variation, thespectrum of P consists of 1 and a subset of {zllzl ~ It < I} for a map like the one weare considering. Eigenvalue 1 comes with multiplicity 1 and the eigenvector h is strictlypositive [8]. Since eigenvectors in 11p are also in [1, we only need to prove that h isin 11p • If this wasn't true, the spectral radius of P in 11p would be less than 1 and wewould have p n l --t 0 in 11p, hence in [', which is impossible. 0
Appendix 2: Combinatorics
We collect here some details on the combinatorical estimates used in the paper.
1. Proof of (23):
Inserting (21) into (22), we get
11181' 1I1~ ~WI-l L e-n::U,iJErHI
r.1"=1'
16
(60)
(61 )
· .
where we used If1- 1 = IYI- 1 ::s; If!.Now, associate to each f in (60) a tree graph r = r(f) with 'F = Y, and write
L=L Lr 7 r:
T(r)=T
and
L li-jl~lrl+ L li-jl(i,;)Er (i.;)Er\T
Finally, the sum over f with r(f) = r is bounded by the sum over all choices of linesconnecting points of Y, which yields:
and this proves (23).
(62)
D
2. Proof of Proposition 4:
Let us first consider the denominator in (35). The basic result of the polymer expansionformalism [3, 21, 24] is that (34) can be written as
whith
L I1( W h)) = exp L Uh)r ~ 'Y
(63)
(64)
where we sum over sequences of polymers in AOn - 1 (not necessarily disjoint or evendistinct) and the sum La is over all connected graphs with vertices {I"", m} andXi; = 0 if ,i n ,; = 0, Xi; = -1 if 'i n ,; # 0, so that this sum vanishes unless, isconnected. Uh) satisfies the bound:
IUh)l::S; 1'8 II (C71)IZIT(Z,2A/3).ZE'Y
(65)
Formulas (63, 64) follow from the polymer formalism, provided we have the bound:
(66)
for any x E Zd+l To prove (66), we use (30) which holds also for I(Wh))I, Slllce11£11 ::s; 1, IIhll ::s; C. Next, note that
(67)
17
, .
where (ZI' ... ,zn) is a sequence of mutually distinct Z; E Zd, and the last sum is over treegraphs on {x, ZI,"', Zn}. TOW, for each fixed tree, the sum over ZI,"', Zn is bounded by(CA-d)n: we start by summing over vertices with incidence number one, remove thosevertices, get a new tree and iterate (i.e., we "roll back" the tree). The number of treegraphs is bounded by cnn! so that (67) is bounded by CTJ.
From this we obtain (66) by repeating the argument in (62), so as to reduce the sum in(66) to a sum over trees whose vertices are now sets Zi E S, and the edges carry powersof Jio. Then the sum over the trees is done as in (67), using the fact that the edges arenow one-dimensional (only in the "time" direction) and that 2:::'=1 Jio" = ~ < 00. Ofcourse we need to choose 7], and therefore c, so that 2; is small enough. This finishesthe proof of (66), hence of (63,64).
To prove (65), we use (64) and sum first over the graphs G corresponding to a giventree as in the proof of (62). Then one has to control the fact that, since, in (64) can bewritten in many ways as a union of ,is, each Z in , can also be decomposed in manyways (each time-like bond in , can also occur several times in the bonds of the ,is, butthis brings extra powers of Jio, and we use 2:::'=1 Jion < 00). Let us write TJ = ~(TJ/~), andT(Z,A)::; T(Z, ~)T(Z, 2;). For any term in (64), we have
and the factor 1, can be absorbed into the constant C in (65). Then, the sum over alln
ways of decomposing, in (64) is bounded by
L(Zi)i=l
UZi = Z
(68)
This in turn is bounded by
(69)
using the fact that the sum (67), and therefore each term in that sum, is bounded by(CTJ); this holds for any A > 0, for TJ small. So we may replace A by ~ and TJ by ~ andobtain (69). This establishes (65).
Now, turning to the numerator in (35), write
p~ = L<,o(r)L II (Wb))r r l1 "'rEf"
(70)
where the sum over r has the same constraints as in Proposition 4, <,o(r) denotes theproduct in (29) (with r v = 0) and the sum over r v runs over sets of disjoint vaccum
18
·.
polymers in AOn- 1 so that 'Yn'Y' = 0, V, E r, V,' E r v . Applying the polymer formalismto this last sum, we get
L II (Wb)) = exp( L Ub)),r .......Er" ..,:
'inr=0
and, using (63), (70) becomes
p~ = LCP(r)exp(- L Ub))r -y:
'inr;"0
(71)
(72)
with r = U~Er 'J. Now, we expand the exponential in (72), and combine each of theterms with r. Concretely, write (72) as
where ,i n r =I 0, Vi. This can be written as
(73)
where the product runs over all polymers (the number of polymers is finite for n and Afinite) and n~ E N. Now, decompose
ru hln~ =I O} = U,:
into mutually disjoint polymers and define I/b'), for a polymer ,', to be given by thesum (73), with the constraint
r u hln~ =I O} = " (74)
Since Ub) is a number, this does not change the type of operators being considered.Since the constraints on the sum over r in (73) are the same as in (36), we have established (36).
The bound (37) follows from (30) applied to the factors in cp(r) and to Ub). The sumover all possible decompositions of " in (74) can be controlled by using (68,69), and bygoing from 2>./3 to >';2. 0
3. Proof of Propositions 5 and 6
The inequalities (45) on eA(nA) follow from (44) and the bounds (37) on III I/b) III, whichhold also for l(vb))I: the polymer formalism allows us to write (44) as exp(L::~ Vb)),which is similar to (63,64) but with 'Y n (Ao U An) =I 0. Then, a bound like (65) implies(45).
19
·.
To prove (46), consider the terms where r :1 ;, with l' n /\0 # 0, l' n /\n # 0. Using(36,37), we bound them by
exp(clAI) 2: JiB II (C7))IZIT(Z, ,\/2)"Y ZE"Y
where eclAI controls the sum over r\; (as in (45)). Next, we extract a factor Jin fromJiB and we control the sum over; by (1 + C7))n as follows: To each;, associate a treeby choosing, for each time t = 1,···, n, a set Z, (possibly empty) and a time-like linejoining successive non empty Z's. The sum over the rest of; is handled as in (62). Thechoice of the lines fixes the "origins", x, of Z,. So, we have IZd choices for each lineand 1/\ I choices for the intersection of; with /\0. These latter factors can be absorbedin ec/AI or Cizi and (1 + C7))n is then an upper bound on
]](1 + z~,(C7))IZIT(Z,~))XtEZ
for any choice of {x,) (as in the bound on (67)). The other terms contributing to theLHS of (46) are bounded in a similar way.
The bound on (53) leading to (47) is also similar. We get, however, ecmin(IXI,1Y1) insteadof eclAI (which is crucial) because here all contributing polymers are "connected" to Xand Y. The connection is defined through the projection of l' on /\, but it is easy tosee that we may again reduce the estimates to a sum over tree graphs and that for twopolymers ;,;' such that 1l',('Y) n 1l',(')") # 0 and 'Yn /\0 # 0,1" n AD # 0, we have that
d('Y','Y) :S 2: [ZJ.ZE"Y'
So, for; fixed, the sum over ;' with 1l',('Y) n 1l',(')") # 0 can be bounded by
2: e-d(x,'Yl 2: JiB II (C7))lzIT(Z, ,\/2)x ;y'3x ZEI"
(75)
where we used (75) to absorb a factor ed{x,'Yl into ITzE"Y' Cizi.Finally, in (55, 56, 57), we perform resummations, using (68,69), which explains why wehave smaller fractions of '\.
20
, .
References
[1] M.L. Blank, Periodicity and periodicity on average in coupled map lattices. Rigorousresults, preprint, Observ. Nice.
[3] D.C. Brydges, A short course on cluster expansions, in: Critical Phenomena, Random Systems, Gauge Theories. Les Houches Session XLIII, K. Osterwalder, R. Storaeds., Elsevier, p.129-183 (1984).
[4] L.A. Bunimovich, E. Carlen, On the problem of stahility in lattice dynamical systems, preprint.
[6] L.A. Bunimovich, Y.G. Sinai, Statistical mechanics of coupled map lattices, in Ref[14].
[7J L.A. Bunimovich, Coupled map lattices: One step forward and two steps back,preprint (1993), to appear in the Proceedings of the "Gran Finale" on Chaos, Orderand Patterns, Como (1993).
[8J P. Collet, Some ergodic properties of maps of the interval, lectures at the CIMPASummer School "Dynamical Systems and Frustrated Systems", to appear.
[9]
[10]
[11]
[12]
[13J
[14]
[15]
[16]
M. Cross, P. Hohenberg, Rev.Mod.Phys. 65, 851-1111 (1993).
N. Dunford, J.T. Schwartz, Linear Operators, Vol. 2, Interscience Publishers, NewYork, p. 1018 (1963).
J-P. Eckmann, D. Ruelle, Ergodic theory of chaos and strange attractors, Rev. Mod.Phys. 57, 617-656 (1985).
J. Glimm, A. Jaffe, T. Spencer, The particle structure of the weakly coupled P(¢)2model and other applications of high temperature expansions, in: ContructiveQuantum Field Theory, G. Velo, A.S. Wightman, eds., Lectures Notes in Physics25, Springer, New York (1973).
J. Glimm, A. Jaffe, Quantum Physics. A functional integral point of view, Springer,
New York (1981).
K. Kaneko (ed): Theory and Applications of Coupled Map Lattices, J. Wiley (1993).
Chaos: Focus Issue on Coupled Map Lattices (ed. K. Kaneko), Chaos 2 (1993).
G. Keller, M. Kiinzle, Transfer Operators for coupled map lattices, Erg. Th. andDyn. Syst. 12, 297-318 (1992).
21
, ,
·.
[17] A. Lasota, M.C. Mackey, Chaos, Fractals and Noise, Springer, New York (1994).
[18] A. Lasota, J. Yorke, On the existence of invariant measures for piecewise monotonictransformations, Trans. Amer. Math. Soc. 186,481-488 (1973).
[19J J. Miller, D.A. Huse, Macroscopic equilibrium from nllcroscopic irreversibility in achaotic coupled-map lattice, Phys. Rev. E, 48, 2528-2535 (1993).
