Random correlations for small perturbations of expanding maps

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Random correlations for small perturbations of expanding maps

BALADI, Viviane, KONDAH, Abdelaziz, SCHMITT, Bernard

BALADI, Viviane, KONDAH, Abdelaziz, SCHMITT, Bernard. Random correlations for small perturbations of expanding maps. Random and Computational Dynamics, 1996, vol. 4, no.

2/3, p. 179-204

Available at:

http://archive-ouverte.unige.ch/unige:12794

Disclaimer: layout of this document may differ from the published version.

Random correlations for small perturbations of expanding maps

Viviane Baladi, Abdelaziz Kondah and Bernard Schmitt

October 1995

Dedicated to the memory of Wieslaw Szlenk

Abstract. We consider random compositions^Fn!F! of^Ck expanding maps^F! which are^Ck-close to a given^Ckexpanding map (^k>1) and not necessarily i.i.d. We study the random correlation functions ^C!(ⁿ) associated to the unique absolutely continuous stationary measures ^F!! =^! and smooth test functions. We show^{Ck ?1} stability of the densities of the measures ^!, and good uniform bounds on the exponential rate of decay of random correlations as the smooth error level goes to zero. To do this, we let the associated random transfer operators ^LF_! act on suitable cones of positive functions endowed with a Hilbert projective metric.

1. Introduction

When studying small random perturbations of a given expanding dynamical system f :X ^!X, i.e., compositions Fⁿ!F! with each random variable F! \close" to f (see Section 2 for precise denitions), one approach is to consider the Markov chain with transition probabilityP(x;E) given by the probability thatF!(x)²E. (See [K1], [K2] for background on random dynamical systems.) The corresponding absolutely continuous invariant measure and its decay of correlations were investigated in [BY]

where strong stability properties were obtained for independent identically distributed (i.i.d.) perturbations of expanding systems by studying an appropriate \integrated"

transfer operator.

In a non i.i.d. setting, there is apparently no adequate integrated transfer operator to study. It is therefore not advantageous a priori to consider integrated correlation functions as in [BY] instead of the random correlations associated to the stationary measures F^j_!^j_! =^j+1_! (see Section 2 for denitions, and also Remark 2.1). Such stationary measures were studied by Kifer ([K3, Theorem B], see also [Ko1, Ko2]), in particular for expanding F!. However, strong stability properties (i.e., convergence results for the densities of these measures) and rates of decay of correlations had not been considered yet. We obtain smooth strong stability of the densities of the stationary measures (Theorem A) uniform exponential decay of random correlations (Theorem B)

1991 Mathematics Subject Classication. 28D20 58F11; 58F30 60G10.

Typeset by ^AMS-TEX

and nally (essentially) optimal upper bounds for these random rates of decay (Theorem C), for expanding systems.

Since we cannot use the integrated transfer operators but need to work with the family of random transfer operators used e.g. by Kifer in [K3], it has proved useful to take advantage of the exibility of the Birkho cones techniques applied to dynamics by Ferrero and Schmitt [FS], and more recently extensively used by Liverani [Li]. This approach has been employed e.g. by Bogenschutz [Bo] in a random framework and should be suitable to study many perturbative situations. We expect the methods in this paper to be applicable with suitable modications to other settings where enough (perhaps nonuniform or piecewise) hyperbolicity is present to guarantee exponential decay of correlations for the original dynamical system f.

Section 2 contains precise denitions and a statement of our results. After recalling some facts about Birkho cones and showing some preliminary bounds in Section 3, we prove Theorem A and Theorem B in Section 4 (using a \naive" family of cones).

In Section 5, introducing \special" cones tailored for our dynamical system, we prove Theorem C. For the reader's convenience, the Appendix contains bounds from [BY].

Instead of rederiving the well-known (see e.g. [Ru]) quasicompactness properties of the transfer operator for f from Theorem B, we could have used them to construct immediately the \special" cones and apply them to the proof of Theorems A and B.

Although this would have made the presentation slightly shorter, we have preferred to give a self contained exposition and use \naive" cones in Sections 3 and 4. (The only exceptions to this \self-contained" rule are the three results from Garrett Birkho stated in Section 3, and ergodicity properties only used in comments; in particular, we do not require the perturbative spectral results from [BY].) This exposition has the advantage of unifying the Hilbert metric and functional analysis approaches, in particular we recover upper bounds for the rate of decay of correlations due to Rychlik ([Ry], see also [Li]).

Work on this paper started during visits of the rst two named authors to the Uni- versite de Bourgogne in Dijon, and we would like to thank the Laboratoire de topologie for its warm hospitality and support. The rst named author is also very grateful to the SFB 170 at the University of Goettingen and to the IHES for their hospitality and nancial support, and acknowledges useful discussions with Thomas Bogenschutz and inspiring questions from Claude-Alain Pillet. Finally, we are very much indebted to Pierre Collet who suggested to us the use of the special family of cones in Section 5.

2. Setting and statement of results

Random compositions and stationary measures.

Let X be a compact, connected, ^C1 Riemannian manifold, and let f : X ^! X be

Cr, with r of the form r = (k;) (meaning that the kth derivative of f is -Holder) for some k ^{2 N}, k 1 and 0 1, with > 0 if k = 1. (We shall say that r⁰ = (k⁰;⁰) r if k⁰ k and ⁰ , and that r⁰ >0 if k⁰+⁰ >0.) We assume that f is locally 0-expanding for some 0 > 1 (i.e., for any x ² X and v ² TxX we have

kDf(x)v^k 0^kv^k). We view the map f as xed throughout; this is the dynamical system which we are randomly perturbing. (We refer e.g. to [KS] for properties of locally expanding maps, recalling only for now that such a map has a constant, nite, number of inverse branches.)

For small > 0 we consider the -neighbourhood B_r(f) of f in the space of all

Cr transformations of X, endowed with the ^Cr metric (i.e., we consider the sum of the ^Ck distance and the C distance between the k^th derivatives). We always assume that is small enough so that all maps in B_r(f) are (locally) expanding. Let be a bimeasurable automorphism of a Lebesgue space (;^D) preserving a probability P, and let F : ^! B(f) be a measurable map. When there is no ambiguity, we shall often write (;) instead of (;), and F_! instead of F(!). We shall consider the random orbits

Fⁿ_!F^n?1_! F_!(x) (2.1) for ! ² (which we view as chosen with P) and x ² X (which we view as chosen with Lebesgue measuredxonX), and let tend to zero. The simplest example is when F() =^ff^g: We are then simply considering our original map. The simplest random example, corresponding to independent, identically distributed perturbations, is when is a two-sided shift space on the symbol setB_r(f), is the full shift,P is a product measure, and F(!) =!0.

Our main tool will be the (random) transfer operators

L!'(x) =^L_F!'(x) = ^X

y²F!^?1(x)

'(y)

jdetDyF!^j (2.2) (see e.g. [K3]) acting on the Banach space of ^Cr?1 test functions ' :X ^{! C} endowed with the ^Cr?1 norm (where r^?1 = (k^?1;)) dened by

kkr^?1 = sup₀

jk^?1sup_X ^kD^j'(x)^k+^jD^k?1'^j;

where^jj denotes the -Holder semi-norm (we write^{k k} for the corresponding norm sup^{j j}+^{j j}) and we have chosen arbitrary norms on the successive tangent spaces.

Observe that ^L_F(dx) = dx for each F ² and small enough , where the ^L_F acts on functionals by duality: (^L_F)(') = (^L_F'). We simply write ^L for the transfer operator ^Lf associated to the map f itself. We may now state our rst main result:

Theorem A (Strong stability of the random measures).

For small enough there is for each ! ² a probability measure on X equivalent with Lebesgue measure !(dx) =h!(x)dx such that F!! =!. Each density h! is the unique ^Cr?1 solution of ^L_!h_! =h_! with ^R h_!(y)dy= 1 and we have

lim^!0_!sup

2^kh!^?h^kr^?1 = 0; (2.3) where h is the unique ^Cr?1 solution of ^Lh=h with ^R h(y)dy = 1.

Clearly, h is the density of an absolutely continuous invariant probability for f. By ergodicity, one may show ([KS]) that hdxis the only such invariant measure for f.

What is new in Theorem A is the strong stability claim (2.3): the existence of the measures h!(x)dxwas proved by Kifer in [K3] in a (similar) Holder setting.

Remark 2.1. Assume for a moment that is a two-sided shift space on the symbol set B_r(f), is the full shift and F(!) = !0. If is ergodic for P, then the measure (dx;d!) = !(dx)P(d!) is an ergodic invariant probability measure for the skew- product T(x;!) = (F;!(x);(!)) on X ([KK, Theorem 3.2]). By the Birkho ergodic theorem is therefore the unique T invariant probability _!⁰(dx)P(d!) with marginal P on whose P-disintegrations _!⁰ are (P-almost all) equivalent with Lebesgue. Writing _Uu for the Dirac mass at a point u ² U, the Birkho theorem also implies that satises ¹_n^Pn_j=0^?1X_Tj(x;!) ^!(dx;d!) and

n1

n^?1

X

j=0^X_Tj(x;!) ^!(dx) :=^Z

!(dx)P(d!)

for dxP! almost all (x;!). Using [K3] it is not very dicult to check thatin the i.i.d.

casethe invariant measure of the Markov chain mentioned in the introduction coincides with the measure (dx) (this is not true in general!).

Correlation functions and random correlation functions.

Writing(dx) =h(x)dxfor the unique absolutely continuousf-invariant probability measure obtained from Theorem A, we introduce the correlation functions for (f;) associated to 1 ²L¹(dx), 2 ²L¹(X) as follows:

C ¹; ²(m) =

Z

X ¹(f^m(x)) 2(x)d(x)^?

Z

X ¹(x)d(x)

Z

X ²(y)d(y); m^2N: (2.4) Note that by ergodicity, we have for Lebesgue almost allx ²X

nlim^!1 1 n

n^X?1

j=0 1(f^j+m(x)) 2(f^j(x)) =^Z

X ¹(f^m(x)) 2(x)d(x): (2.5) It is well known that under our assumptions theC(n) decay exponentially fast (see e.g.

[Ry, Ru]). We shall in fact recover this result as a consequence of Theorem B below, with the help of a useful and standard rewriting of (2.4):

C ¹_; ²(m) =^Z

X ¹(x)

Lm( 2h)(x)dx^?h(x)^?Z 2(y)h(y)dydx; m ^2N: (2.6) For small enoughand each! ², dene for 1 ²L¹(dx), 2²L¹(X) and allm0 the random correlation

C

1; ²;!(m) =^Z

X ¹(F^m! F^m?1!F!(x)) 2(x)d!(x)

? Z

X ¹(x)d^m+1_!(x) ^Z

X ²(y)d_!(y) (2.7)

(where the measures ! are given by Theorem A).

Remark 2.2. In the setting of Remark 2.1, we get for Lebesgue almost all x ² X and P almost all !

nlim^!1 1 n

n^X?1

j=0 1(T^j+m(x;!)) 2(T_j(x;!)) =

Z

X 1(T^m(x;!)) 2(x)!(dx)P(d!): (2.8) (Note in particular that 1(T_n(x;!) depends on !.) The object of interest therefore appears to be

C ¹_; ²(m) =^Z

C ¹_; ²_;!(m)P(d!): (2.9) However, except in the i.i.d. case where the integrated correlation C ¹_; ²(n) can be expressed in terms of the Markov chain and its invariant measure (see [BY]) and studied with the help of an integrated transfer operator, the functionC ¹_; ² does not seem easier to handle than the random correlation (2.7).

The second main result follows:

Theorem B (Uniform bounds for the random rates of mixing).

Let r⁰ = (k⁰;⁰)>0 with r⁰ r^?1. For small enough and for each ! ² there is _;!;r⁰ <1 and a constant K >0 so that for all 1 ²L¹(dx), 2 ^2Cr⁰(X), and all n0

jC

1; ²;!(n)^jK _n;!;r^{0?Z j} 1(x)^jdx^k 2^kr⁰: (2.10) There is a constant ~_r⁰ < 1 such that

limsup

!0 _!sup

2_;!;r⁰ ~_r⁰: (2.11)

Quasicompactness of the transfer operator.

We dene for each 0 < r⁰ r ^? 1, each suciently small and each ! ² , the exponential rate of decay _;!;r⁰ of the random correlation C_! in ^Cr0(X) to be the smallest number ;!;r⁰ so that for any ;!;r⁰ > ;!;r⁰ (2.10) holds for some constant K > 0 (depending perhaps on ;!;r⁰=;!;r⁰) and all 1 ²L¹(dx), 2 ^2Cr⁰(X), n0.

Applying Theorem B to the case where F() = ^ff^g, and using (2.6) for suitable

1 >0 with^R 1(x)dx= 1, we see that for all 0< r⁰ r^?1

xsup²X

Ln'(x)^?h(x)^Z '(y)dy

K ~_nr^0k'^k_r⁰; ⁸'^2Cr0(X);n0: (2.12) Since ^Lmh = h, using < 1 from the Yorke inequality in Lemma 4.2 below (and the fact that sup_X^Ln1 is uniformly bounded by (4.2), and therefore ^kLnk_r⁰ is uniformly bounded) we nd a constant ~K so that

kL2n'^?h^Z '(y)dy^k_r⁰ K^~ max(~_nr⁰;ⁿ)^k'^k_r⁰; ⁸'^2Cr0(X);n0: (2.13)

Therefore, for any 0 < r⁰ r^?1 the spectral radius of ^L acting on ^Cr0(X) is equal to one, the only eigenvalue on the unit circle is the simple eigenvalue 1, and there are no other elements of the spectrum of modulus greater than max(~r⁰;)¹⁼² (in particular, the operator is quasicompact). Let us denote for 0< r⁰ r^?1

r⁰ := sup^fjz^jjz ²spectrum(^L:^Cr0(X)^!Cr0(X));z⁶= 1^g (2.14) then_r⁰ max(~_r⁰;)¹⁼² <1 by (2.6), and for any > _r⁰, there is a constant K()>0 so that for any 1; 2 ^2Cr⁰(X)

jC ¹; ²(n)^jK ^{n?Z j} 1^jdx^k 2^kr^{0 8}n0: (2.15) Note that, whenever r⁰ is the modulus of an eigenvalue of ^L, it is the smallest number so that (2.15) holds for all > r⁰. (Use (2.6) for 2 a corresponding eigenfunction and for suitable ^R 1 = 1.)

Finally, recalling that limm^!1(^Lm1)^1=m 1 by (4.2), we dene r⁰ = limsup_m

!1

xsup²X

X

y²f^?m(x)

kD(f_y^?m)(x)^k(k0+0)

jdetDf^m(y)^j

1=m

limm(^Lm1)^1=m

⁽₀^k0+0) <1: (2.16) (for y² f^?m(x) we letf_y^?m be the corresponding local inverse branch in a neighbour- hood of x).

Theorem C (Good bounds for the random rates of mixing).

For0 < r⁰ r^?1 let ;!;r⁰ be the exponential rate of decay of the random correlation C_! in ^Cr0(X) and let r⁰, r⁰ be as dened in (2.14), (2.16). Then

limsup

!0 _!sup

2;!;r⁰ max(r⁰;r⁰): (2.17) In dimension one, it is not dicult to adapt the methods in [CI, BJL] to show that when 0< r⁰ r^?1 the essential spectral radius^r_e⁰_ss(^L) of ^Lacting on^Cr0(X) coincides with r⁰, so that in particular max(r⁰;r⁰) = r⁰. It follows that Theorem C gives optimal upper bounds in dimension one (recall that [CI] also prove that any 0< t < r⁰

is an eigenvalue of ^L).

We conjecture that the property r⁰ = ^r_ess⁰ (^L) (and therefore max(r⁰;r⁰) = r⁰) holds in higher dimensions too for r⁰ r^?1.

We alsoconjecture that, if we assume further that _r⁰ > ^r_ess⁰ (^L) for some r⁰ r^?1,

then lim

!0

?ess sup_!²_;!;r⁰ =_r⁰: (2.18) Finally, we would like to point out that our results do not depend on the choice of the probability measure P: what matters is that the random variables considered lie in the ball B_r(f).

3. Transfer operators acting on function cones and Hilbert metrics

Transfer operators acting on cones.

First assume that r = (1;) for some 0 < 1. In this case, we work with the following family of convex cones _L = _L;, indexed by L >0:

L =^f'^2C0(X)^j'(x)>0;⁸x²X ; '⁽x)

'(y) e^Ld(x;y); ⁸x;y ²X^g: (3.1) As usual in applying projective methods, the rst step consists in showing that our operators map suitable cones strictly inside themselves (recall that 0 > 1 is the expansion constant of f):

Lemma 3.1.

Assume that r = (1;) with 0< 1. Fix some 1 > > 1=₀. There are 0 >0 and L0 <¹, so that for any < 0, any F ²B¹⁺(f) and all L > L0

LFL L: (3.2)

(Bounds similar to Lemma 3.1 have been obtained in many settings, see e.g. [Li].) Proof of Lemma 3.1. We start by observing thatf satises the following stability prop- erty: let us x some 0 > > 1 (assuming for further use that > 1=) then there is 0 > 0 such that for any F ² B¹⁺() with < 0 F is locally -expanding (in particular infX^jdetDF^{j d} where d is the dimension of X). It follows that for any x;y ² X the sets F^?1(x) and F^?1(y) are in bijection in such a way that the distance d(x⁰;y⁰) between two paired points is not greater than d(x;y)=. (First prove this for d(x;y) suciently small and then use the fact that the metric on X comes from the Riemannian structure.) Moreover, by compactness, the -Holder constants of the jaco- bians^jdetDF^jfor such F are uniformly bounded above by someQ >0. It follows that for any F in B¹⁺(f) and any x⁰;y^{0 2}X we have (using suplog⁰(u)1=infu)

jdetDF(x⁰)^j

jdetDF(y⁰)^j =e^{logjdetDF(x0)j?logjdetDF(y0)j}

e^{(jdetDF(x0)j?jdetDF(y0)j)=d} e^Qd(x0;y0)=d: ^(3.3) Fix L > 0 and ' in L. Then for any F in B¹⁺(f) and any x;y ² X we get by applying (3.3) and using the pairs (x⁰;y⁰(x⁰))²F^?1(x)F^?1(y) described above:

LF'(x) ^X

x⁰²F^?1(x)

'(y⁰)e^Ld(x0;y0)

jdetDF(x⁰)^j

X

x⁰²F^?1(x)

'(y⁰)e^Ld(x0;y0)e^Qd(x0;y0)=d

jdetDF(y⁰)^j

LF'(y)e^{(L+Q=d)(d(x;y)=)}:

(3.4)

Since > , it suces to take L0 = (Q=^d+)=(^{? 1}).

Consider now the caser = (k;) with k 2, assuming for simplicity that = 0 (the modications needed for noninteger exponents above 2 are mostly left to the reader).

We now dene ⁰_{L; ~M} = ⁰_{L; ~M;k}?1 for L >0 and ~M = (M1 >0;::: ;Mk^?1 >0) by ⁰_{L; ~M} =^f'^2Cr?1(X)^j'(x)>0;⁸x²M ; '⁽x)

'(y) e^Ld(x;y);⁸x;y²X

kD^j'(x)^kMj'(x);⁸x²M ;1j k^?1^g: ^(3.5) (When > 0, we add the condition that D^k?1' is -Holder with Holder constant bounded by some M.)

Lemma 3.2.

Let r = (k;0) with k 2 and x some 1 > >1=0. There are 0 > 0, L⁰₀ > 0, ^M0;1 > 0 and functions ^M0;j : ^Rj₊^{?1 ! R}+ for 2 j k ^?1, so that for < 0, any F ² B_r(f), and L > L⁰₀, Mj > ^M0;j(M1;::: ;Mj^?1) (1 j k^?1) we have

LF(⁰_{L; ~M})⁰_L;~M; (3.6)

where we write ~M = (M1;::: ;M_k^?1).

Proof of Lemma 3.2. We may x some 0 > > 1 (so that > 1=) and assume that is small enough so that all maps F in B_r(f) are -expanding, and that the jth derivatives, 1 j k^?1, of their jacobian ^jdetDF^j and also the norms of the jth derivatives D^jF for 1 j k are bounded by some uniform constant Q > 0. In particular, (3.3) and (3.4) hold for = 1, and since >1=, it suces to take

L⁰₀ Q=^d+1

^{? 1} ; (3.7)

to ensure that the second condition on the cones is satised.

Let us now write in detail the calculations for the rst two derivatives. For any F ²B_r(f), any '²⁰_{L; ~M} and any x²X we have

kD^LF'(x)^{k X}

F(y)=x

kD'(y)^k

^jdetDF(y)^j + ^X

F(y)=x

'(y)^kD^jdetDF(y)^{jk j}detDF(y)^j2

X

F(y)=x

M1'(y)

^jdetDF(y)^j + ^X

F(y)=x

'(y)^kD^jdetDF(y)^{jk (1+d)j}detDF(y)^j

M1

⁺ Q

^(1+d)

LF'(x):

(3.8)

We may thus take

M0;1 = (Q=^(1+d))=(^?1=): (3.9)

For k 3 we have (using >1 and d 1)

kD^2LF'(x)^{k X}

F(y)=x

kD²'(y)^k

^2jdetDF(y)^j + ^X

F(y)=x

3Q^kD'(y)^{k 2j}detDF(y)^j

+ ^X

F(y)=x

(Q+ 3Q²)'(y) ^3jdetDF(y)^j

M2

^{2 + 3}QM1

^{2 +} Q+ 3Q^{2 3}

LF'(x):

(3.10)

We may thus take

M0;2(M1) ((Q+ 3Q²)=+ 3QM1)=^{2 ? 12} :

For the general casek 4, we use that we may bound^kD<sup>`L</sup>G'(x)<sup>k</sup>with 3`k^?1 similarly as in (3.10), where each term on the right-hand side is either bounded by

X

F(y)=x

pj(Q)^kD^j'(y)^{k `j}detDF(y)^j

X

F(y)=x

pj(Q)Mj'(y)

<sup>`j</sup>detDF(y)<sup>j</sup> ; (3.11) with 1j `^?1 and p_j() a polynomial (depending on), or by

X

F(y)=x

p0(Q)'(y)

^`jdetDF(y)^j; (3.12)

where p0() is a polynomial, except a single term of the form

X

F(y)=x

kD^{`</sup>'(y)<sup>k `j}detDF(y)^j

X

F(y)=x

M`'(y)

^`jdetDF(y)^j: (3.13)

Projective metrics in vector lattices.

To proceed, we need to recall basic denitions and results about projective metrics on positive cones in vector lattices (see [Bi2], and also [N] for a more recent exposition;

[L] contains a lucid and short account). Let ^E be a topological vector space and a continuous partial ordering (i.e., if ;'n ^2E, 'nfor alln, and limn^!1'n='then

'). We call such a pair anintegrally closed vector lattice. A subset ^E is called a coneif'² for all'² and all^2R₊; the cone is calledclosedif ^[f0^gis closed.

We dene the Hilbert pseudo-metric on the positive cone =^f'^{2 Enf}0^gj0 '^g as follows: for '; ², set

('; ) = sup^{f2R+ j}' ^g; ('; ) = inf^{f2R+ j} '^g; (3.14) (where we dene = 0 and =¹ when the corresponding sets are empty), then let

('; ) = log ('; )

('; ): (3.15)

Our main tool will be:

Birkho's inequality ([Bi1], [Bi2]).

If ^L:^{E !E} is a linear map from the integrally closed vector lattice ^E into itself such that ^L() for the corresponding positive cone, then for any '; ² we have

(^L';^L )tanh

diam(^L) 4

('; ): (3.16)

(Where we use the notation diam(A) = sup_'; ²_A('; ).)

We shall need two additional results from Birkho in the special case where (^E = C⁰(X);) is an integrally closed vector lattice structure on the vector spaceC⁰(X) of complex valued continuous functions on a compact metric space X (endowed with the topology of uniform convergence), with the positive cone (;) having the property that'(x)>0 for all '², and where we are given a Borel probability measure dmon X with full support:

Hilbert and uniform metrics ([Bi1]).

For '; ² with ^R_X'dm=^R_X dm= 1, we have

sup_X ^j'(x)^? (x)^j(e^{('; )?}1)sup_X '(x): (3.17)

Completeness Lemma ([Bi1]).

For any 0 ² with ^R 0dm= 1 the set

f'^{2j Z}

X'dm= 1; ('; 0)<^1g (3.18) is a complete metric space for the metric .

The next step is to dene the vector lattices which we shall use. We shall always take ^E to be ^C0(X) with the uniform metric, and dm = dx the normalised Lebesgue measure on X. Consider rst the case r = (1;) where 0 < 1. Fixing > ₀ and L > L0() as given by Lemma 3.1, we observe that L as in (3.1) is a closed convex cone in C⁰(X) and that _L^\?_L =^;. We dene the order _L by:

'L ^{$ ?}'²L^[f0^g: (3.19)

One proves that that the positive cone associated with L coincides with L. A standard computation (see e.g. [FS; Ko1; Li, Lemma 2.2]) shows that, for any'; ²_L:

('; ) = inf_x

6=y²X e^Ld(x;y) (x)^? (y) e^Ld(x;y)'(x)^?'(y) _xinf

2X (x) '(x); and ('; ) = sup_x

6=y²X e^Ld(x;y) (x)^? (y)

e^Ld(x;y)'(x)^?'(y) _xsup

2X (x) '(x);

so that

_L;('; ) = log sup_x

6=y²X u⁶=v²X

(e^Ld(x;y) (x)^? (y))(e^Ld(u;v)'(u)^?'(v))

(e^Ld(x;y)'(x)^?'(y))(e^Ld(u;v) (u)^? (v)): (3.20) Consider now integerr2. Similarly, we may dene an order ⁰_{L; ~M} using the cones ⁰_{L; ~M} in (3.5). We then get for '²⁰_{L; ~M}

⁰(';1) = min

1(';1); _xinf

2X 1jk^?1

Mj

Mj'(x) +^kD^j'(x)^k

; and ⁰(';1) = max

1(';1); _xsup

2X 1jk^?1

Mj

M_j'(x)^?kD^j'(x)^k

; so that

⁰_{L; ~M;k}^?₁(';1) = max

L;1(';1);log sup_x;y

2X 1j;`k^?1

M`

M_j Mj'(x) +^kD^j'(x)^k M<sub>`</sub>'(y)<sup>?k</sup>D<sup>`</sup>'(y)^k; log sup_x

6=y²Xu²X 1jk^?1

(e^Ld(x;y)?1)(M_j'(u) +^kD^j'(u)^k) (e^Ld(x;y)'(x)^?'(y))Mj ; log sup_x

2X u⁶=v²X 1`k^?1

M<sub>`</sub>(e<sup>Ld(u;v)</sup>'(u)<sup>?</sup>'(v)) (M`'(x)^?kD^`'(x)^k)(e^Ld(u;v)?1)

:

(3.21)

Clearly, the next step is to get uniform bounds for the diameter of^L_FL in _L (see Lemma 2.3 in [Li] for a very similar bound), respectively ^LF⁰_{L; ~M} in ⁰_{L; ~M}:

Lemma 3.3.

(1) For any L >0, <1

diamL;_L 2log 1 +

1^{? + 2}LdiamX =:K(;L): (3.22) (2) For any r = (k;0) with k 2, and any L, ~M, <1

diam⁰_{L; ~M;k}?1

⁰_{L; ~M;k}?1 K(;L): (3.23)

Proof of Lemma 3.3.

(1) We use formula (3.20) for '; ²_L and get L;('; )log sup_x

6=y²X u⁶=v²X

(e^{Ld(x;y) ?}e^?Ld(x;y))(e^{Ld(u;v) ?}e^?Ld(u;v)) (y)'(v) (e^{Ld(x;y) ?}e^Ld(x;y))(e^{Ld(u;v) ?}e^Ld(u;v))'(y) (v)

log

(1 +)²

(1^?)² e^2LdiamX:

(3.24) (2) We now use (3.21) which involves the maximum of four expressions. The rst one was dealt with in (1) (noting that each 2 may be replaced by 1 in the right-hand-side of (3.24) when = 1). For the second, we clearly have for '²⁰_{L; ~M}:

log sup_x;y

2X 1`;jk^?1

M`

M_j Mj(1 +)'(x) M_`(1^?)'(y) log

1 +

1^? e^LdiamX

: (3.25)

The third and fourth expressions are quite similar. We only consider the third one, and see that for any '²⁰_{L; ~M}

log sup_x

6=y²Xu²X 1jk^?1

(e^Ld(x;y)?1)(M_j'(u) +^kD^j'(u)^k) (e^Ld(x;y)'(x)^?'(y))Mj

log sup_x

6=y²X²Xu²X 1jk^?1

(e^Ld(x;y)?1)(1 +)M_j'(u) (e^Ld(x;y)?e^Ld(x;y))Mj'(x)

log

1 +

1^?e^LdiamX

:

(3.26)

4. Stability of random measures and bounds for the decay of random correlations

The following lemma is quite standard in our setting:

Lemma 4.1.

Consider the case r = (1;) with 0 < 1, and let L0 and be as in Lemma 3.1. For all suciently small and all L > L0 there is a uniquely dened map h: ^!L (we write h! for h(!) as usual) such that for each ! ²

Z

Xh_!(x)dx= 1 and ^L_!h_! =h_!: (4.1)