·
~
-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 dimensional 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 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 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
Cf:l~"(-CT93-0411
1
The first rigorous results on such systems are due to Bunimovich and Sinai who
studied a one dimensional lattice of weakly coupled maps [5]. They established the
existence 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 Gibbs
state for the corresponding two-dimensional spin system. This is a natural extension of
the method used for a single map or for hyperbolic systems [25, 231. These results were
strengthened by Volevich [27] and extended by Pesin and Sinai to coupled hyperbolic
attractors [20] (for a review, see [6]). An extension to lattices of any dimension is
announced in [28] (for coupled hyperbolic attractors).
Since the Gibbs measure constructed by Bunimovich and Sinai describes statistical
mechanics in two dimensions, the possibility of phase transitions i.e. non-uniqueness of
invariant measure is open (for recent results on this, see [1, 2, 4, 7, 19, 22] and references
therein). In statistical mechanics, Gibbs measures are often easy to construct in weak
coupling (which corresponds to high temperature) and in strong coupling (low temperature) using convergent expansions. The purpose of the present paper is to develop these
expansion methods for the dynamical system problems in infinite dimensions.
We consider weakly coupled circle maps and derive a convergent cluster expansion for
the Perron-Frobenius operator (transfer matrix in the statistical mechanics terminology).
This allows us to prove exponential mixing in space and time for an invariant SRB
measure. These results are similar to those of Bunimovich and Sinai, but our method
works immediately in any dimension and is simpler. However, for technical reasons we
need to restrict ourselves to real analytic maps.
The Perron-Frobenius operator has been a powerful tool to analyze quite general
maps, of bounded variation [18] (for reviews see [8, 17]). This approach was used also
for coupled maps in [16], but weaker results were obtained there in the infinite volume
limit. An open problem still remains to develop expansion methods for coupled maps
that are of bounded variation. These are the most natural candidates that might exhibit
interesting phase transitions as the coupling is increased.
We have tried to make the paper self-contained for readers having no background
in the expansion methods of statistical mechanics. Appendix 2 contains some of the
standard combinatorical estimates needed. A reader who is familiar with these methods
will find a slightly novel application of them because our expansion is applied directly
to the Perron- Frobeni us operator.
2
Results
We consider the following infinite dimensional dynamical system. The state space of the
system is
M = (8 1)Zd,
the direct product of circles over Zd, i.e., m EM is given as m = {m;}ieZd, mi E 8 1.
M carries the product topology and the Borel a-algebra inherited from 8 1 . To describe
the 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
parameter. We define now the coupled map T : M ---+ M by
E
> 0 is a
(3)
T=<lio:F.
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 such
that gn are analytic in V x V and
(4)
for some C <
00,
Let us denote by
A > 0 and all u, v Eli.
0";
the shifts in Zd, i = 1, ... , d. Then our main results are
Theorem. Suppose f and gn satisfy A, B. There is an
there exists a B01'el meaSU1'e It on }\II such that
Eo
>0
such that for
E
<
Eo
1. p, is invariant unde1' T and the shifts 0";. F01' any finite A C Zd, the ma1'ginal
distribution of p, on (5 1)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. Tn m
---+
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 are
usually called "space-time chaos".
Remark 2. The result holds for a much more general class of interactions <Ii : we could
take
<Ii(m); = m;e2~;Lx'iexgX(mx)
where X C Zd, mx
the bound
= mix, IXI is finite and
gx : (5 1 )IX 1
---+
R is analytic in Vixi with
Igx(zx)1 ~ E1X1e-h(X)
where r( X) is the length of the shortest tree graph on the set X. Then the theorem holds
again for E < Eo small enough provided gx +j = gx, j E Zd Without this translation
invariance It is not O";-invariant, but the other claims still hold. Similarily, F may be
replaced by x ;Ezdf; where f; satisfy A, uniformly in i. Our results also extend to coupled
maps of the interval [0,1], of the type considered by Bunimovich and Sinai [5], provided
we take their f and Q analytic.
Remark 3. We do not prove, but conjecture, that there is only one invariant measure
whose local marginal distributions on the sets (5 1)A, A C Zd,IAI finite are absolutely
3
'-.
-'
continuous with respect to Lebesgue measure. This is true in a class of measures with
analytic marginals: the proof of 3 extends to the case where m is replaced with a measure
satisfying some clustering and the analyticity of the densities of the local marginal
distributions. This is similar to the results of Volevich [27,28]. Thus JL can be considered
as 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 a
nearest neighbour in A, or if A consists of a single point). We denote M A = (5 1)A Let
now 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 T A = <I> A 0 IA invariant measure JLA on J\lh and later JL as a suitable
limit of JLA. The measures JLA will be constructed by means of the Perron-Frobenius
operator of T A • To describe this, wel1rst collect some straightforward facts about I and
its invariant measure.
The Perron-Frobenius operator P for
J
(g
0
I on L 1 (5') is defined as usual by
J)h dm =
J
gPh dm
for hE L 1 (5 1 ),g E Loo(5 1 ) and dm the Lebesgue measure (i.e., dx in the parametrization
m = e2..iX ). We wish to consider P on a smaller space, namely H p , the space of bounded
holomorphic functions on the annulus A p = {1- P < Izi < 1 + p}. H p is a Banach space
in the sup norm. The assumption A for I implies the following spectral properties for
P:
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
(5)
with Q a i-dimensional projection opemto''';
Qg= (Lgdm)h=e(g)h
with h E Hpo,h > 0 on 5',
lSI
III
for some JL < 1, C <
00
R
(6)
hdm = 1 and
n
III:::; CJL n ,
and all n; we use
QR = RQ = 0
III . III
(7)
to denote opemtol' norms.
Remark 1. a) is a consequence of the expansiveness of I and shows that the operator
P i~proves. the domain of analyticity, while b) means that there is a unique absolutely
contlOuous IOvarlant measure for I, with density in H po ' and the rest of the spectrum of
·
~
Pin ff po 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 change
from place to place, even in the same equation.
To describe the Perron-Frobenius operator P A for the coupled map T A , we introduce
some notation. We denote 'H.~ = 0 iEA H p , P A = 0 iEA P i where Pi acts on the i-th
variable. Also, we denote by dmA the Lebesgue measure on M A. Then
(8)
where
(ili~G)(m) = detDili;;-'(m)G(ili;;-l(m))
(9)
and thus
(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, 13J
for iliA in terms of localized operators with good bounds on norms. Then we shall
construct the invariant measure in Section 4 by studying the limit n --> 00 of Ph; a
cluster 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 Jollowing
holds 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~ =
Ly 0yEYOY 0
(11)
1A\y
whe,'e Y runs though sets oj disjoint subsets oj A, A \ Y = A \ uY and 1z denotes the
imbedding 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 linear
order 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 > 0
such that all g,,'s are holomorphic in ApI x API ( A p 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 the
following
Lemma. Thel'e exists c;o(Pl, I', A) sitch that, JOT c; < C;o(p" I', A) <I>;;:~ is a holomoTphic
Jamily (JOI'S E D/I.) oj holomoTphic diffeomoTphisms JTOm A~,_s into A~, wheTe 0 -+ 0
as 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
(16)
is the nm'm in 'H~,_S'
Proof. (4) and (15) imply
(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) follows
0
from Hadamard's inequality [10J.
Thus, <I>A. : 'H~,
-+ 'H~,-S
The cluster expansion for
theorem of calculus:
is a halomorphic family of operators, for sED /I..
<I>~
is the following repeated application of the fundamental
(19)
where the notation is as follows. r is a subset of Ax A and each pair (i,j) E r is
r
IT(i,j)Er ds ij ,
= IT(i,j)Er a~i)' Sr< = {Sij} (i,j)Er< and Irl is the
such that i < j. ds r
cardinality of r. (19) then follows by writing 1 = ITi<j(Iij + E ij ) with I ij = f[olJ dSija~i)
and E ij is the evaluation map at Sij = 0 (for more details, see [12] and Proposition 18.2.2
in [13]).
Now observe that OAr factorizes as follows. For a set A C A x A, let A denote the
union of its projections on the two factors. We say PI, pz E r are connected to each
other if 15, n 15z =I 0. Let rex be the maximal connected components of r with respect to
this relation. Then we have
=
a
(20)
6
·"
where "['ofo
fbyf a ).
:
Hr;
-t
Hr~-s is given by the integral in (19) (with A replaced by fa and
We now estimate the norm of "['of o ' 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) is
inside 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 connected
graph on the points of Y, pick a connected tree graph T with T = Y and estimate the
sum over f in (22) by
III"FIlI ::; ( L
e-t'\ljl)IYI~1Y1-I
jEZd
where ~
,. - t 00
L e-~ITI
(23)
T
r- 1 exp[Cc:r(l + A-d)] (see Appendix 2 for more details). Since we may take
as c: - t 0, see (16-18), the claim (12) follows.
D
=
Remark. Propositions 1 and 2 imply that there is an C:o > 0 and p > 0 such that P A
maps H~ into itself for all A and all c: ::; C:o: the domain of analyticity shrinks by an
amount" 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 and
for all.
4
Space time expansion for the invariant measure
The TA -invariant measure ItA will be constructed by studying P:\ as n
yield spectral information on P A uniformly in A. From (8), we have
- t 00.
This will
(24)
into which we insert (11) for <Pi. and, for PA, we use (5):
PA = (/9iEA (Qi
+ R;) =
L (/9;EI R; (/9jEA\I
leA
Qj
==
L R[ (/9 QA\I'
(25)
leA
and we get
P:\ =
L L
n
II((/9FEY, "F
{I,) (Y,) '=1
7
(/91 A\y,)(R 1, (/9 QA\I,)
(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 some
finite 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
(27)
where we used Q2
=Q
repeatedly. Let us write Q in (6) as
Q= hr2;£
(28)
where we identify bounded operators Iip ---+ H p with elements of H p tensored with its
dual H; and use' to distinguish this tensoring from the one used for spaces indexed by
Zd. Using the shorthand
£1' == I8!;El' £, Iw = l8!iEY h
and inserting (28) into (27), the latter becomes
HI 18! (h A\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 to
establish a correspondence between the terms in (26) and geometrical objects defined
on this lattice.
We let S denote the set of all finite "spacelike" subsets of Zd+l, i.e. Z E S is of the
form Z = Y x {t} for some Y C Zd and t E Z. Also, let B denote the set of "timelike
bonds" of Zd+l, i.e. & E B is of the form & = {(i, t), (i, t + 1)} == &;(t) for some i E Zd
and integer t. The correspondence with terms in (26) is defined as follows: to each
Y E Yt, t = 1,···, n, we associate Z = Y x {t} and to each It we associate the set of
bonds {&;(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 , to
be 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 respect
to this relation, or if , consists of a single element belonging to S. Denote by "I the
support of " i.e. the subset of Zd+1 that is the union of the elements of ,. We say
that 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 first
and the second factor. Then 71't('f) is connected, i.e. it is an interval denoted [L, t+]. Let
8
,
.
..
'Y± = 7r s (;yn 7r,l(t±)) The weight of the polymer "I, W("(), is a bounded linear operator
W("() : H;- -> H;+ or equivalently W("() E H;+0(H;-)*. W in the example above is a
weight 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 t 2 • In (26) we will encounter a family of
polymers f in AOn and we need to make a distinction whether their support intersects
the boundary of Aon i.e. Ao or An. We denote the family of those intersecting neither by
f v (v stands for "vacuum polymers"), those intersecting Ao but not An by f o, the ones
intersecting 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 exactly
one 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 between
the points of Y. On the other hand, bonds are associated to R factors and long strings
of such factors are suppressed by (7). Note however that J.L need not be small, only less
than one. Thus, this expansion is similar to the one of a lattice of weakly coupled one
dimensional systems, but where, within each system, the couplings are not necessarily
small.
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.,er n W("()h)0(eL
(29)
0.,ero ew("()))
where the sum is ove,' sets f (possibly empty) oj mutually disjoint polymers "I with
fJ"( C AoUA n . The Jollowing notation was used: ew("() = e.,+ W("() E (H;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
t,
III
W("()
111:5 J.LB II (C7))IZIT(Z,A)
(30)
Ze.,
III . III
where B is the number oj bonds in "I and
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) liE
1,+, , t E [O,n -In and {Y,};'::l a subset of S, So = {Y x {t} , Y E Y,}. Decompose
8 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
P A= 'L0r(A)
(31 )
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
identity
r
and Ao U An by the identity operator. For the remaining Q's use the
QKQ = e(K h)Q,
(32)
where K : ti p ---> 7i p , 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) where
the weights W(-y) : 7i;- ---> 7i;+ are given by (here F E 7i;-)
(33)
Yin (33) is 1r s ('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 linear
functionals is also denoted by II . II) and the fact that on bonds b of Y x 1r,('7) that are
disjoint from '7, we have identity operators.
0
Equation (29) is an example of polymer expansion in statistical mechanics and it is well
known (see e.g. [3, 21, 24]) that the bounds (30) will enable us to prove exponential
falloff of correlations (i.e. mixing) and construct the A ---> Zd limit (in the standard
treatments the weights of polymers are scalars and not operators as here, but, as we will
see, the combinatorical part of the proof is as in the standard case). We refer the reader
not familiar with the combinatorical methods needed in this analysis to the references
cited 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 us
call these the vacuum polymers. These are "freely floating" in AOn - 1 unlike the others
that are attached to the boundary Ao U An, but they tend to cancel each other. To
see this, note that, by (10) with G = 1, A pk = A for all k, so, since e(h) = 1, and
Rh = RQh = 0 (which implies W(-y)h = 0 for 'Y E f o U fOn), we get
e
1
e
= eA(p~-lhA) = L:
r
(34)
mW(-y))
.,
where f is a set of disjoint vacuum polymers in AOn - 1 (note that the vacuum polymers
in (29) lie in AOn - 1 too). The cancellation we are after is accomplished by using (34) to
write
p~ =
(L:
r
mW(-y)))-lp~
.,
(35)
and substituting (29). The standard combinatorics (see Appendix 2) now yields
Proposition 4. (35) can be written as
p~
= L:@"EronV(-y)@ ((h A+ @.,Er n V(-y)h)0(eAr
@"ErOeV(-Y)))
when the sum is over sets f (possibly empty) oj disjoint polymers 'Y in Aon with
(A o U An) # 0 and 8'Y C Ao U An. The weights satisJy
III
V(-y)
III:::; JiB
II (C,/)lzIT(Z,A/2).
ZE.,
10
(36)
'7 n
(37)
Notice now that, as n --t 00, the I E f On in (36) give exponentially small contributions
due to (37), and (36) will factorize. Let us define a function DA E HZ
DA= :L>'Alr+ I8i..,Er IIb)h
(38)
I'
where f is a set of finite, mutually disjoint polymers I in A- ooo with "I n Ao oF
Eh C Ao. Similarily, define a linear functional £A : HZ --t R by
£A
=
0 and
L eAlr _18i"'Er ellb)
(39)
r
where I are now in Aooo , with 0 oF Elf
c
Ao. Put
(40)
Then we have
Proposition 5. (Spectml decomposition oj PA). There exist co > 0, 1-'-1 < 1, c <
independent oj A, such that, JOI' c < co, nA and LA have the Jollowing properties
Iln AII ::; eclA1 , LA = eA
PAn A= nA
III P~ - nA~LA III ::; I'~
00,
(41 )
e
clAI
(42)
(43)
J01' all A C Zd and n EN.
Remark. Thus the spectrum of P A consists of the eigenvalue 1 with multiplicity 1 and
the rest is in the disc {izi ::; I'd, uniformly in A.
Proof. We have
eA(D A) =
and (using
Ilhll ::;
C)
L II (lib))
I'
II DAII::; C IAI L
I'
Since
II ell::;
(44)
..,
II.., II lib) II .
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 canceled
by 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) and
again 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,
o
and then use the continuity of P A .
We will pass shortly to the 1\ = Zd limit but before that we need a more refined mixing
condition than (43):
Proposition 6. Thel'e exists PI < 1, c < 00 such that, if dpA = fJAdmA denotes
the T A invariant meaSUl'e constructed in Proposition 5, then, for any X, Y c 1\ and
F E 1i xp ,G E 1i yp
,
Proof. We want to compare
(48)
with
fA(FOA)fA(GO A).
To do this, we expand them. Let us denote the terms in the sum (38) by
ones in (39) by lA(r). Then we get, see (40,'11),
fA(FO A) = lA(Fn A) = I:lA(r)FnA(f')
n (f) and the
A
(49)
ff'
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 under
this 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(f y )Fny(f~)
fy,r;..
Y:XcY
12
(50)
Since £z(D. z ) = 1 for all Z, (50) equals
L
£A(FD. A) =
lIF (r)
(51 )
['
where f is a set of polymers I in A- ooo or in Aooo , intersecting Ao and with projections
on 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 (f 2 )
rt nr2=0
+L
lIF ,a(f)
(52)
r
where f = U~El' "1, the first sum comes from the f On = 0 terms in (36) and, in the second
one, 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 7f s b) are now
connected 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 an
expansion like (52) but with f 1 n f 2 i 0 in the first sum and then each term in both
sums contributes fln e-+d(X,V) (as in the proof of (43)). The combinatorics is controlled
D
by e,min(IXI,1Y1l because each f must be connected to X and Y (see Appendix 2).
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 the
invariant measure fl. To do this, rewrite A in (38) as
n
fi A
=
L
@~ Vb)h A
['
with Vb)
and thus
= V(~)h,
liVll
'+
.
Since h > 0 on Sl (Proposition 1), h
i
0 on Sp for p small enough
satisfies (37) too. Thus we may exponentiate (see Appendix 2)
fi A
=
exp(LUb))h A ,
..,
(54)
U satisfies
(55)
Define, for YeA, H y E 'H.~ by
Ifl' =
L
,)'+=y
13
Ub)
<
•
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 Hamiltonian
H, for TJ small. We take f.L to be this measure, i.e., the unique Borel probability measure
on M with conditional probability densities given by
f.L (d mA ImAO ) =
exp[- Ll'nA;f0 H}'(m)] I d
'A mA
J 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 translation
invariant. For the T-invariance of f.L it suffices to show
J F
0
T df.L = J F df.L
H;,
for all F E
all X finite. Let T(A) = T A 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 sup
norm. (56) and (57) imply that I,(A) -> f.L weakly (here A -> Zd is taken in the sense of
the 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 exist
C < 00, such that for all finite X C Zd, all F, G E H;, and all n E Zd+l
a> 0,
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
n_oo
0
Tndm = JFdf.L
14
·.
But,
and
So, it is enough to show
with Ill, c, independent of A. But this follows from (47) with G = hI!'
D
Acknowledgments
We would like to thank L.A. Bunimovich, E. Jarvenpaa, J. Losson and Y.G. Sinai
for interesting discussions. This work was supported by NSF grant DMS-9205296 and
by 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
Thus, our assumptions are: There exists a Po > 0, 1 > 1 such that
I
again.
(a) I is holomorphic and bounded on 5 po = {z E C1IIrnzl < Po} with I(R) = R,
(b) I(z
+ 1) = I(z) + k z E 5 po '
(c)J'(X)~,
kEN, k ~ 2,
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 5 po and
5'"!po respectively. Let Ii p be the space of periodic bounded holomorphic functions on
5 p , of period one (which can be identified with Iip of Sect.2). Since P is given by
(Fc )(z) =
g
I: 1'(,p(z+J))
(1* + j))
g
(59)
j=O
we get from the above remarks that P : Iip -> Ii'"!p for p ::::; Po, which is the claim a) of
Proposition 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
n.
L
l,pq(z
+ j) -,pq(jWlIg(n)llp/2
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 ~ ,-q
in 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 such
that k q 2- n < ~. Then
III
pq -
Pqn
111< ~.
From this we conclude, since Pq" is of finite rank, that the spectrum of pq outside of
the 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 the
other hand it is well known that in a space of ['-functions of bounded variation, the
spectrum of P consists of 1 and a subset of {zllzl ~ It < I} for a map like the one we
are considering. Eigenvalue 1 comes with multiplicity 1 and the eigenvector h is strictly
positive [8]. Since eigenvectors in 11p are also in [1, we only need to prove that h is
in 11p • If this wasn't true, the spectral radius of P in 11p would be less than 1 and we
would 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,iJEr HI
r.
1"=1'
16
(60)
·.
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
L
r
7
r:
T(r)=T
and
L
li-jl~lrl+
(i,;)Er
L
li-jl
(61 )
(i.;)Er\T
Finally, the sum over f with r(f) = r is bounded by the sum over all choices of lines
connecting points of Y, which yields:
(62)
and this proves (23).
D
2. Proof of Proposition 4:
Let us first consider the denominator in (35). The basic result of the polymer expansion
formalism [3, 21, 24] is that (34) can be written as
L
r
I1( W h)) = exp L Uh)
~
(63)
'Y
whith
(64)
where we sum over sequences of polymers in AOn - 1 (not necessarily disjoint or even
distinct) and the sum La is over all connected graphs with vertices {I"", m} and
Xi; = 0 if ,i n ,; = 0, Xi; = -1 if
n ,; # 0, so that this sum vanishes unless, is
connected. Uh) satisfies the bound:
'i
IUh)l::S; 1'8 II (C71)IZIT(Z,2A/3).
(65)
ZE'Y
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
11£11 ::s; 1, IIhll ::s; C. Next, note that
I(Wh))I,
Slllce
(67)
17
,
.
where (ZI' ... ,zn) is a sequence of mutually distinct Z; E Zd, and the last sum is over tree
graphs on {x, ZI,"', Zn}. TOW, for each fixed tree, the sum over ZI,"', Zn is bounded by
(C A-d)n: we start by summing over vertices with incidence number one, remove those
vertices, get a new tree and iterate (i.e., we "roll back" the tree). The number of tree
graphs 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 powers
of Jio. Then the sum over the trees is done as in (67), using the fact that the edges are
now one-dimensional (only in the "time" direction) and that 2:::'=1 Jio" = ~ < 00. Of
course we need to choose 7], and therefore c, so that 2; is small enough. This finishes
the proof of (66), hence of (63,64).
To prove (65), we use (64) and sum first over the graphs G corresponding to a given
tree as in the proof of (62). Then one has to control the fact that, since, in (64) can be
written in many ways as a union of ,is, each Z in , can also be decomposed in many
ways (each time-like bond in , can also occur several times in the bonds of the ,is, but
this brings extra powers of Jio, and we use 2:::'=1 Jion < 00). Let us write TJ = ~(TJ/~), and
T(Z,A)::; T(Z, ~)T(Z, 2;). For any term in (64), we have
and the factor 1,n can be absorbed into the constant C in (65). Then, the sum over all
ways of decomposing, in (64) is bounded by
(68)
L
(Zi)i=l
=Z
UZi
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 ~ and
obtain (69). This establishes (65).
Now, turning to the numerator in (35), write
p~
= L<,o(r)L
r
r
l1
II (Wb))
(70)
"'rEf"
where the sum over r has the same constraints as in Proposition 4, <,o(r) denotes the
product 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'
to this last sum, we get
= 0, V, E r, V,' E r v . Applying the polymer formalism
L II (Wb)) = exp( L
r .......Er"
Ub)),
(71)
Ub))
(72)
..,:
'inr=0
and, using (63), (70) becomes
L
p~ = LCP(r)exp(r
-y:
'inr;"0
r
= U~Er 'J. Now, we expand the exponential in (72), and combine each of the
with
terms 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 A
finite) and n~ E N. Now, decompose
ru
=I O}
hln~
into mutually disjoint polymers and define
sum (73), with the constraint
= U,:
I/b'), for
r u hln~ =I O}
= "
a polymer ,', to be given by the
(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 sum
over all possible decompositions of " in (74) can be controlled by using (68,69), and by
going from 2>./3 to >';2.
0
3. Proof of Propositions 5 and 6
The inequalities (45) on eA(n A) follow from (44) and the bounds (37) on III I/b) III, which
hold 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 (A o U An) =I 0. Then, a bound like (65) implies
(45).
19
·.
To prove (46), consider the terms where
(36,37), we bound them by
exp(clAI)
r
:1 ;, with
l' n /\0 # 0, l' n /\n # 0. Using
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 from
JiB and we control the sum over; by (1 + C7))n as follows: To each;, associate a tree
by choosing, for each time t = 1,···, n, a set Z, (possibly empty) and a time-like line
joining successive non empty Z's. The sum over the rest of; is handled as in (62). The
choice of the lines fixes the "origins", x, of Z,. So, we have IZd choices for each line
and 1/\I choices for the intersection of; with /\0. These latter factors can be absorbed
in 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 the
LHS of (46) are bounded in a similar way.
The bound on (53) leading to (47) is also similar. We get, however, ecmin (IXI,1Y1) instead
of eclAI (which is crucial) because here all contributing polymers are "connected" to X
and Y. The connection is defined through the projection of l' on /\, but it is easy to
see that we may again reduce the estimates to a sum over tree graphs and that for two
polymers ;,;' such that 1l',('Y) n 1l',(')") # 0 and 'Yn /\0 # 0,1" n AD # 0, we have that
d('Y','Y) :S
2: [ZJ.
(75)
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"
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 we
have smaller fractions of '\.
20
,
.
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22