LINEAR RESPONSE FUNCTION FOR
COUPLED HYPERBOLIC ATTRACTORS
MIAOHUA JIANG AND RAFAEL DE LA LLAVE
ABSTRACT
We prove that when we take the thermodynamic limit in the context of coupled
hyperbolic attractors, Ruelle’s derivative formula of the SRB measure with respect
to the underlying dynamical system remains true if one of the terms is interpreted
appropriately.
1. Introduction: Derivative formulas of the SRB measure and its
entropy
For uniformly hyperbolic systems, Ruelle proved that the (generalized) SRB measure depends on the system differentiably and calculated its derivative formula [14, 16].
The derivative formula of the SRB state with respect to the underlying system is also
called the linear response function and is used to develop the theory of non-equilibrium
statistical mechanics [15]. Since Statistical Mechanics is mainly concerned with infinite
systems and the relevant physical quantities are obtained through thermodynamic limits, it is of interest to study the thermodynamic limit of the linear response function.
The differentiability of the thermodynamic limit of SRB measures was established in
[7]. In this paper, we want to study the thermodynamic limit of the formulas for the
derivative of the SRB measures.
For a finite dimensional hyperbolic attractor f , let ϕ(x) be a differentiable function
on M and δf be the vector field on M in which the derivative of the SRB measure µf
is taken. Denote X = δf ◦ f −1 . Ruelle’s derivative formula states that
∞ Z
X
[< grad(ϕ(f n )), X s > −ϕ(f n )divuσ X u ]dµf ,
(1.1)
δµf (ϕ) =
n=0
where X s and X u are projections of X onto the stable and unstable subspaces, respectively, and divuσ X u is the divergence of the vector field X u on the unstable manifold
with respect to the canonical metric σ induced by the SRB measure. Even though
the vector field X u is in general, not differentiable as observed by D. Dolgopyat, its
1
2
M. JIANG AND R. DE LA LLAVE
divergence is well-defined and it is Hölder continuous on the hyperbolic attractor [16].
The formula (1.1) is expected to hold whenever the SRB measure exists and is
differentiable with respect to the map f [17]. In this paper, combining the techniques
developed in our proof [7] of the smooth dependence of the SRB measure for coupled
hyperbolic systems and the strategy that Ruelle used to obtain the formula in finite
dimensional systems [14, 16], we prove that an analogous formula for the derivative
holds in the thermodynamic limit for coupled hyperbolic attractors. We just need to
require that the function ϕ(x) is sufficiently differentiable and that it satisfies certain
decay property commonly needed for infinite-dimensional systems. We also need to give
a proper interpretation to the term involving divergences: While the divergence divuσ X u
is infinite for most meaningful vector fields on the coupled system, the divergence of
the projection of each component of X onto the unstable subspace Xiu can be shown
to be finite and
∞ Z
X
ϕ(Φn )divuσ X u dµ
n=0
can be interpreted as
∞ XZ
∞ Z
X
X
u u
n
ϕ(Φn )divuσ Xiu dµ.
ϕ(Φ )divσ X dµ =
n=0 i∈Zd
n=0
The sum on the righthand side converges because of the exponential decay of correlation
functions. Indeed, we will also show that the sum
∞ Z
X
(1.2)
ϕ(Φn )divuσ XVu dµV
n=0
converges as V → Zd , where ΦV , XV denote the restrictions of the coupled map Φ and
the vector filed X onto the finite product MV = ⊗i∈V M , V ⊂ Zd is a finite set. As
usual in statistical mechanics, we will need to understand that the limit V → Zd is
reached in the sense of van Hove [19]. We will carry out the details only when V are
cubes but indicate the easy modifications to get the general statement. We denote by
µV is the SRB measure of ΦV . Thus, we can use the limit as V → Zd in (1.2) as the
definition of the series
∞ Z
X
ϕ(Φn )divuσ X u dµ.
n=0
LINEAR RESPONSE FUNCTION
3
We then, have the same derivative formula for coupled hyperbolic attractors
∞ Z
X
δµΦ (ψ) =
[< grad(ϕ(Φn )), X s > −ϕ(Φn )divuσ X u ]dµΦ .
n=0
In the finite dimensional case, the derivative formula of the entropy with respect to
the SRB measure is given by [16]
∞ Z
X
δhµf (f ) =
− log J u f (f n (x))divuσ X u dµg .
n=−∞
By an analogous argument, the derivative formula of the entropy in the thermodynamic
limit is given by
∞ Z
X
δhµΦ (Φ) =
[− log Jf u + ψ ] ◦ Φn (x̄)divuσ X u dµΦ ,
n=−∞
where − log Jf u + ψ is the potential function for the SRB measure µΦ [10, 7] and the
integral
Z
[− log Jf u + ψ ] ◦ Φn (x̄)divuσ X u dµΦ
is defined via the thermodynamic limit of finite dimensional approximation as V → Zd .
For additional recent results in the area of smooth dependence and the differentiation
of SRB measures, we refer readers to [1], [5], [12], and [18].
2. Notation and Formulation of the Main Result
First, we briefly introduce the notation. For more details, see [7].
1. Let M be a smooth compact Riemannian manifold and f a C r (r > 4) diffeomorphism of M . Assume that f possesses a hyperbolic attractor Λf . The direct product
space ⊗i∈Zd Mi over the integer lattice of dimension n is denoted by M, where Mi is a
copy of M . This direct product space is a Banach manifold under the usual supremum
metric ρ
ρ(x̄, ȳ) = sup d(xi , yi ), x̄ = (xi ), ȳ = (yi ) ∈ M.
i∈Zd
M is also equipped with the so called “compact metric” ρq , 0 < q < 1.
(2.1)
ρq (x̄, ȳ) = sup q |i| d(xi , yi ), x̄ = (xi ), ȳ = (yi ) ∈ M,
i∈Zd
4
M. JIANG AND R. DE LA LLAVE
P
where i = (i1 , i2 , · · · , id ) ∈ Zd and |i| is the lenght, |i| = dk=1 |ik |. Note that, the
supremun metric is invariant under translations of the indices i, but the compact
metric is not.
The infinite dimensional system F = ⊗i∈Zd f over M is called the uncoupled system
and it possesses a hyperbolic attractor ∆ = ⊗i∈Zd Λf .
2. For the integer lattice Zd , given α0 > d, θ0 ≥ 0, there exists a > 0 such that the
function defined by
(
a|i|−α0 exp(−θ0 |i|) i 6= 0,
(2.2)
Γ(i) =
a
i = 0,
satisfies the following properties:
P
(1) i∈Zd Γ(i) < ∞,
P
(2) j∈Zd Γ(i − j)Γ(j − k) ≤ Γ(i − k).
A function satisfying these two conditions is called a decay function. (See [7] Proposition 3.2).
In this paper, we will only consider a fixed decay function of the form (2.2) and with
θ0 > 0. Let Vn = {i : i ∈ Zd , |i| ≤ n} and Vbn = Zd \ Vn . The following properties can
be easily verified for decay functions as in (2.2).
(1) For any number α > 0, there exist some constants C and β
depending only on the decay function Γ and α, such that
X
(Γ(i))α ≤ C exp(−βn).
i∈Vbn
(2) For any number α > 0, there exist some constants C and β
depending only on the decay function Γ and α, such that
X
(Γ(i − j)Γ(j − k))α ≤ C exp(−β|i − k|).
j∈Zd
Throughout the paper, we will use C, α, and β to denote generic constants that
will appear in various definitions and estimates. Instead of using different symbols for
different constants, we will use these generic symbols with explicit explanations of their
dependence.
3. A coupled map lattice Φ is defined to be a small perturbation of F of the form
Φ = G ◦ F,
LINEAR RESPONSE FUNCTION
5
where G is a C r , r > 4 diffeomorphism of M close to the identity. By lifting G(x̄) to
the tangent space Tx̄ M, we can identify G with a smooth section of the tangent bundle
T M.
4. Now we give the definitions of Banach spaces for the objects we study.
(1) C 0 (M, T M), the space of continuous sections of the tangent bundle:
{v̄(x̄) : x̄ → v(x̄) ∈ Tx̄ M is continuous, ||v̄||C 0 ≡ sup sup |vi (x̄)| < ∞}
i∈Zd x̄∈M
(2)C α (M, T M), the space of Hölder continuous sections of the tangent bundle:
For 0 < α < 1, we fix a trivialization of T M by a finite number of coordinate charts,
{P1 , . . . , Pl }
(2.3)
C α (M, T M) =
{v̄(x̄) : ||v̄||C α ≡ max(||v̄||C 0 , sup max
i∈Zd
k
sup
x̄6=ȳ∈M,xi ,yi ∈Pk
|vi (x̄) − vi (ȳ)|
< ∞)}
d(x̄, ȳ)α
The Banach space of all Hölder continuous functions on M with Hölder exponent α
will be denoted by C α (M, R).
(3) CΓα (M, T M), the space of Hölder continuous sections satisfying decay property
specified by Γ:
Let h : M → Rn be a function. Define
γα,i (h) = sup
sup
(xj )j6=i (zj =yj =xj )j6=i ,yi 6=zi
kh(ȳ) − h(z̄)k
,
dα (yi , zi )
where the first supremum is taken over all x̄ = (xj ) without the ith variable and the
second supremum is taken over all pairs of ȳ and z̄ with their variables equal to that
of x̄ except at the lattice site i.
We can identify vj , the jth variable of v = (vj ), with a function from M to Rn . For
0 < α <≤ 1.
CΓα (M, T M) =
{v ∈ C 0 (M, T M) : kvkCΓα ≡ max{kvkC 0 , sup γα,j (vi )Γ−1 (i − j)} < ∞}.
i,j∈Zd
(4) CΓr (M, T M), the space of C r -sections satisfying decay property specified by Γ:
6
M. JIANG AND R. DE LA LLAVE
For each r ∈ N, we denote
CΓr (M, T M) = v ∈ C r (M, T M) ∩ CΓ1 (M, T M) :
∂k
kvkCΓr ≡ max sup sup
vi (x̄)C 0 max{Γ−1 (i − i1 ), · · · Γ−1 (i − ik )}
0≤k≤r i ,··· ,i ∈Zd i∈Zd ∂xi1 · · · ∂xik
1
k
<∞ .
The space of C r functions on M satisfying decay property specified by Γ will be
denoted by CΓr (M, R). For any function ϕ ∈ CΓr (M, R), we have
∂ϕ ∂xi ≤ Γ(i)kϕkCΓr .
For convenience, we also introduce a semi-norm for each coupled map
kΦkCΓr = max
sup
sup
0≤k≤r i ,··· ,i ∈Zd i∈Zd
1
k
∂k
Φi (x̄)C 0 max{Γ−1 (i−i1 ), · · · , Γ−1 (i−ik )}
∂xi1 · · · ∂xik
This norm is finite as long as the CΓr -norm of the corresponding perturbation function
G is finite.
The following results on the existence, uniqueness, and the differentiability of the
SRB measure for the coupled map lattice Φ are contained in [7, 10].
Given a decay function Γ, a smooth Riemannian manifold M , and a C r (r > 4)
system f with a topologically mixing hyperbolic attractor Λ, there exists a CΓr neighborhood U of the uncoupled system F = ⊗i∈Zd f such that for all Φ ∈ U, the following
statements are true.
1) There is a topologically mixing hyperbolic attractor set ∆Φ close
to ∆F .
2) There is a map hΦ : ∆F → ∆Φ such that
Φ ◦ hΦ = hΦ ◦ F.
The map hΦ is unique among those that are CΓ0 close to the
identity. It is CΓα for some α > 0.
3) The map Φ → hΦ is C r−2 considered as a map from CΓr to CΓα .
4) There exists a measure µΦ on ∆Φ (called SRB measure), invariant under Φ, such that it is the limit of the SRB measures
of finite approximations of the coupled lattice map.
LINEAR RESPONSE FUNCTION
7
5) Given any function ϕ ∈ CΓr−1 (M, R), the function Aϕ : U → R
defined by
Z
Aϕ (Φ) =
ϕ dµΦ
is C r−3 when U is given the CΓr topology.
In this article, we obtain the derivative formula for the function
Z
AΨ (Φ) = ϕ dµΦ
under the conditions of the previous theorem.
Theorem 1. Let δf ∈ CΓr (M, T M) be a C r -vector (r > 4) field on M. Assume that
Φ = G ◦ F ∈ U with G ∈ CΓr (M, T M) and µΦ is its SRB measure. Let X = δf (Φ−1 ).
Then, for any function ϕ(x̄) ∈ CΓr−1 (M, R), the functional
Z
ϕ → ϕdµΦ
is differentiable with respect to Φ = G ◦ F, G ∈ CΓr (M, T M) in the direction of δf .
The derivative (linear response function) is given by
∞ Z
X
[< grad(ϕ(Φn )), X s > −ϕ(Φn )divuσ X u ]dµΦ ,
< δµΦ , ϕ >=
n=0
R
where the integral ϕ(Φn )divuσ X u is defined by taking the limit in the finite dimensional
approximation as V → Zd in the sense of van Hove.
3. The proof
3.1. The strategy of the proof.
We use the following standard approach to obtain the derivative:
Proposition 2. Assume that gV () is a sequence of real-valued functions and converges
to g() as V → Zd for every ∈ (−0 , 0 ). Assume that gV () is C 1 in and gV0 () is
bounded uniformly in V and that limV →Zd gV0 () = g̃() in (−0 , 0 ). Then, g() is C 1
and g 0 () = g̃().
The differentiability of the SRB measure with respect to Φ was proved in [7] using
symbolic representation of the system. Here, in the process of determining the derivative formula, we obtain another proof based on Ruelle’s derivative formula of the SRB
measures for finite dimensional systems. This proof gives the derivative formula in the
infinite dimensional setting as the limit of the derivative formulas of finite dimensional
approximation. The proof in [7] established existence of the derivative for hyperbolic
8
M. JIANG AND R. DE LA LLAVE
sets but did not provide a formula. The present paper establishes the formula for attractors and also proves the convergence of the finite dimensional approximations for
the derivative.
3.2. Finite dimensional approximations. In the rest of the section, we make precise the notions of the finite approximations to the coupled map lattice.
Pick any point x̄∗ ∈ ∆Φ , which we will use as the boundary conditions at infinity.
It will be a byproduct of our work that one obtains the same results no matter which
point we choose.
Given any set V ⊂ Zd – it suffices to consider the case that V is a cube –, we define
the map ΦV : xV → ΦV (xV ) on MV = ⊗i∈V Mi is the restriction of Φ on MV .
∗ ΦV (xV ) = Φ(xV , xVb ) .
V
The coordinatewise description is given by
ΦV (xV ) i = Φ(xV , x∗Vb ) i ,
i ∈ V,
where the point (xV , x∗Vb ) denotes an element in M whose restrictions to V and its
complement Vb ≡ Zd − V , are xV and x∗Vb , respectively.
The map ΦV is a diffeomorphism of MV when the perturbation G is sufficiently close
to the identity and it is C 1 -close to the diffeomorphism FV = ⊗i∈V f . By the structural
stability theorem, ΦV possesses a hyperbolic attractor ∆ΦV since FV has a hyperbolic
attractor ∆FV = ⊗i∈V Λ. There exists a conjugating homeomorphism hV : ∆FV → ∆ΦV ,
ΦV ◦ hV = hV ◦ FV .
We note that, the hyperbolicity constants are uniform in V . Hence, we can apply the
structural stability theorem for all V . See [7] for more details.
For any smooth function ϕ ∈ CΓr−1 (M, R), we can similarly define its restriction to
the finite product MV :
ϕV (xV ) = ϕ(xV , x∗Vb ).
It is clearly a C r−1 -function on MV with respect to the differentiable structure induced
by the direct product of Riemannian metric.
We fix a coupled map Φ = G ◦ F in U. Let δf ∈ CΓr (M, T M) be a translation
invariant vector field. We describe the -perturbation of Φ in the direction of δf by
the map
Φ (x̄) = A−1
Φ(x̄) δf (Φ(x̄)),
LINEAR RESPONSE FUNCTION
9
where belongs to certain open interval (−0 , 0 ) and AΦ(x̄) is the exponential map at
Φ(x̄).
We need to truncate the map Φ over the finite volume V :
∗ ΦV, (xV ) = Φ (xV , xVb ) .
V
Note that
∗
Φ (xV , x∗Vb ) = [A−1
δf
(Φ(x
,
x
))]
.
V
Φ(xV ,x∗ )
Vb
V
V
b
V
So, it is an -perturbation of the map ΦV in the direction of the vector field
∗∗ δfV = δf (xV , xVb ) ,
V
where x∗∗ = Φ(xV , x∗Vb ).
With V fixed, we apply the Ruelle’s derivative formula (1.1) to the finite dimensional
system (ΦV, , MV ) in the direction of δfV .
The function
Z
gV () : →
ϕV dµΦV,
∆ΦV,
is C 1 in and the derivative is given by
∞ Z
dgV () X
=
[< grad(ϕV (ΦnV, )), XVs > −ϕ(ΦnV, )divu XVu ]dµΦV, ,
d
n=0 ∆ΦV,
where XVs and XVu are projections of the vector field XV = δf (Φ−1
V, ) onto the stable
and unstable subspaces of DΦV, and ∆ΦV, is the hyperbolic attractor for ΦV, .
Since we already know that
Z
Z
lim gV () = lim
ϕV dµΦV, = g() =:
V →Zd
V →Zd
∆ΦV,
ϕdµΦ ,
∆ Φ
in order to obtain the derivative formula for the SRB measure of Φ, we only need to
prove the following statements on the derivative of gV ():
Proposition 3. (1) As functions of , both infinite series
∞ Z
X
< grad(ϕV (ΦnV, )), XVs > dµΦV,
n=0
and
∞ Z
X
(ϕ ◦ ΦnV, )divu XVu dµΦV,
n=0
are uniformly bounded in V .
10
M. JIANG AND R. DE LA LLAVE
(2) For each ∈ (−0 , 0 ), the following limits exist.
(a)
∞ Z
∞ Z
X
X
s
n
lim
< grad(ϕV (ΦV, )), XV > dµΦV, =
< grad(ϕ(Φn )), X s > dµΦ .
V →Zd
n=0
n=0
(b)
lim
V
→Zd
∞ Z
X
ϕ(ΦnV, )divu XVu dµΦV,
n=0
=
∞ Z
X
ϕ(Φn )divu X u dµΦ .
n=0
These limits should be considered as the definitions of the righthand sides.
3.3. Proof of Proposition 3: Convergence of the first term. In this section, we
consider the convergence of:
∞ Z
X
(3.1)
< grad(ϕV (ΦnV, )), XVs > dµΦV, .
n=0
3.3.1. Uniform boundedness in V . We first estimate the integral
Z
< gradϕV (ΦnV, ), XVs > dµΦV, .
For any function ϕ ∈ CΓr−1 (M, R), its gradient is well defined. Assume that X ∈
C 0 (M, T M) is a bounded vector field on M. We have the following useful estimates.
The proofs are omitted as the estimates follow directly from the decay property of ϕ.
The norm kXk denotes the C 0 -norm of the vector field X and kϕk denotes the CΓr−1
norm of ϕ.
Lemma 1. For any ϕ ∈ CΓr−1 (M, R) and X ∈ C 0 (M, T M), we have
(1)
X
Γ(i).
| < gradϕ, X > | ≤ kϕkkXk
i∈Zd
(2) If X = ⊕i∈Zd Xi , Xi ∈ T Mi and |Xi | ≤ CΓ(i − j) for some constant C, then
| < gradϕ, X > | ≤ CkϕkkXkΓ(j).
We also notice that
< gradϕV (ΦnV, ), XVs >=< gradϕV , DΦnV, XVs >
and by the uniform contraction condition
kDΦnV, XVs k ≤ C 0 λn kXVs k,
LINEAR RESPONSE FUNCTION
11
where C 0 > 0 is a constant and 0 < λ < 1 is the contracting constant along the stable
direction. Note that as long as our 0 is chosen small, the contracting constant λ can
be so chosen that it is independent of and V . Thus, by the decay property of ϕV and
Lemma 1 (1) we have
X
| < gradϕV , DΦnV, XVs > | ≤ CC 0 λn kϕkkXVs k
Γ(i).
i∈Zd
So, as functions of ,
∞ Z
X
< grad(ϕV (ΦnV, )), XVs > dµΦV, ≤
n=0
X
C
kϕkkXk
Γ(i)
1−λ
d
i∈Z
is uniformly bounded in V .
3.3.2. Convergence as V → Zd . Because of the uniform convergence of the series
∞ Z
X
< grad(ϕV (ΦnV, )), XVs > dµΦV,
n=0
in V , to prove that its limit as V → Zd is
∞ Z
X
< grad(ϕ(Φn )), X s > dµΦ ,
n=0
we just need to show that for each fixed n,
Z
Z
n
s
lim
< grad(ϕV (ΦV, )), XV > dµΦV, =
< grad(ϕ(Φn )), X s > dµΦ .
V →Zd
s
Let XV,i
, Xis be the ith coordinate of the vector fields XVs , X s , respectively. We have
X
X
s
XVs =
XV,i
, Xs =
Xis .
i∈V
i∈Zd
By Lemma 1, the term < gradϕ(Φn ), X s > is well-defined and
X
< gradϕ(Φn ), X s >=
< gradϕ(Φn ), Xis > .
i∈Zd
Since
< gradϕV (ΦnV, ), XVs >=
X
i∈V
s
< gradϕV (ΦnV, ), XV,i
>
12
M. JIANG AND R. DE LA LLAVE
and the measure µΦ converges to µΦ weakly, it suffices to show that for each fixed n
and all V
X
s
| < gradϕV (ΦnV, ), XV,i
> | ≤ ai , and
ai < ∞
i∈Zd
and for each i
s
lim < gradϕV (ΦnV, ), XV,i
>=< gradϕ(Φn ), Xis >
V →Zd
uniformly in x̄ ∈ M.
We first prove several properties needed in the proof of the convergence. For simplicity, we drop the subscript CΓr in the norm kΦkCΓr : kΦk = kΦkCΓr .
Lemma 2.
(1) For each n, there exists a constant C(n) depending only on the norm kΦk and the
size of the manifold M such that the ith coordinates of ΦnV, and Φn satisfy
X
d( ΦnV, (xV ) i , Φn (x̄) i ) ≤ C(n)
Γ(i − j)
j∈Vb
for every i ∈ V and x̄ ∈ M.
(2) For each n, the truncated function ϕV (ΦnV, (xV )) converges to ϕ(Φn (x̄)) uniformly
in x̄ and . Indeed, we have
X
|ϕV (ΦnV, (xV )) − ϕ(Φn (x̄))| ≤ (C(n) + kΦk)kϕk
Γ(j)
j∈Vb
(3) For each i ∈ Zd and n ∈ N,
as V → Zd .
(4) For each fixed n, we have
∂
ϕ (ΦnV, (xV ))
∂xi V
converges to
∂
ϕ(Φn )
∂xi
exponentially
< gradϕ(ΦnV ), Xi > | ≤ kXkkϕkkΦkn Γ(i).
Proof. (1) Notice that for the ith coordinate, i ∈ V , we have
d( Φ,V (xV ) i , Φ (x̄) i ) = d( Φ (xV , x∗Vb ) i , Φ (x̄) i )
X ∂Φ,i
≤
|
|d(x∗j , xj ),
∂xj
j∈Vb
x∗j
where
denotes the coordinate of x∗ at j. Let L be the maximum distance between
two points on the compact manifold M . Note that
∂Φ,i
|
| ≤ kΦkΓ(i − j).
∂xj
LINEAR RESPONSE FUNCTION
13
We have
X
d( ΦV, (xV ) i , Φ (x̄) i ) ≤ LkΦk
Γ(i − j).
j∈Vb
In general, for the ith coordinate, i ∈ V and ȳ = (yi )i∈Zd = (yV , x∗Vb ), we have
d( ΦV, (yV ) i , Φ (x̄) i ) = d( Φ (yV , x∗Vb ) i , Φ (x̄) i )
X ∂Φ,i
|d(yj , xj )
≤
|
∂xj
j∈Zd
X
≤ LkΦk
Γ(i − j)d(yj , xj ).
j∈Zd
Thus,
X ∂Φ,i
d( Φ2V, (xV ) i , Φ2 (x̄) i ) = d( ΦV, (yV ) i , Φ (z̄) i ) ≤
d(yj , zj ),
∂xj
d
j∈Z
where ȳ = (ΦV, (xV ), x∗Vb ) and z̄ = (zj )j∈Zd = Φ (x̄). Using the previous estimation, we
have:
d( Φ2V, (xV ) i , Φ2 (x̄) i )
X
X
≤ LkΦk
Γ(i − j)d(yj , zj ) + LkΦk
Γ(i − j)d(x∗j , zj )
j∈V
≤ LkΦk
X
j∈Vb
Γ(i − j)LkΦk
j∈V
X
Γ(j − k) + L2 kΦk
k∈Vb
≤ L2 kΦk2
XX
≤ L2 kΦk2
X
k∈Vb j∈V
X
Γ(i − j)
j∈Vb
Γ(i − k) + L2 kΦk
k∈Vb
X
Γ(i − j)
j∈Vb
X
Γ(i − j)
j∈Vb
Γ(i − j)Γ(j − k) + L2 kΦk
≤ L2 (kΦk2 + kΦk)
X
Γ(i − j).
j∈Vb
By induction in n, we have, for xV = x̄|V ,
X
| ΦnV, (xV ) i − Φn (x̄) i | ≤ C(n)
Γ(i − j),
j∈Vb
where C(n) is a constant depending on L, kΦk and n.
14
M. JIANG AND R. DE LA LLAVE
The claim (2) is now a direct consequence of (1).
|ϕ(ΦnV, (xV ), x∗Vb ) − ϕ(Φn (x̄))|
X ∂ϕ
|d( ΦnV, (xV ) i , Φn (x̄) i )
≤
|
∂xi
i∈Zd
X
X
X
≤ C(n)kϕk
Γ(i)
Γ(i − j) + Lkϕk
Γ(i)
i∈V
≤ kϕk(C(n) + L)
j∈Vb
X
i∈Vb
Γ(i).
i∈Vb
(3) With i ∈ V and n fixed, we wish to estimate
∂
∂
∂
∂
n
n
n
∗
n
=
.
ϕ
(Φ
(x
))
−
ϕ(Φ
(x̄))
ϕ(Φ
(x
),
x
)
−
ϕ(Φ
(x̄))
b
V
V,
V
V,
V
∂xi
∂xi
V
∂xi
∂xi
We may drop the symbol in our calculation since the conditions involved are always
independent of .
Using the chain rule, we have:
(3.2)
∂
∂
ϕ(ΦnV (xV ), x∗Vb ) −
ϕ(Φn (x̄))
∂xi
∂xi
X ∂ϕ ∂(ΦnV )j
∂ϕ ∂(Φn )j ≤
−
,
∂xj (ΦnV (xV ),x∗Vb ) ∂xi
∂xj Φn (x̄) ∂xi
d
j∈Z
where (ΦnV )j and (Φn )j denote the jth coordinate of maps ΦnV and Φn , respectively.
We note that since the manifold M is multidimensional the products in the formula
are actually matrix products.
We first estimate the following difference between partial derivatives of ϕ. We use
the fact that the decay function Γ is of the form (2.2) and the elementary inequality
LINEAR RESPONSE FUNCTION
15
1
min{a, b} ≤ (ab) 2 for a, b ≥ 0.
∂ϕ ∂ϕ −
∂xj (ΦnV (xV ),xV∗b ) ∂xj Φn (x̄)
X ∂2ϕ n
∗
n
≤
∂xj ∂xk d((ΦV (xV ), xVb ), Φ (x̄))
k∈Zd
≤
(3.3)
X
kϕk min{Γ(j), Γ(k)}C(n)
k∈Zd
X
Γ(k − j)
j∈Vb
1
≤ kϕkC(n)Γ(j) 2
X
k∈Zd
1
≤ kϕkC(n)Γ(j) 2
1
Γ(k) 2
XX
X
Γ(k − j)
j∈Vb
1
Γ(k) 2 Γ(k − j)
k∈Zd j∈Vb
1
2
≤ kϕkC(n)Γ(j) e−βl(V ) ,
where β is some constant and l(V ) is the linear size of V . Note that for all j ∈ Zd , we
also have
∂ϕ
| ≤ kϕkΓ(j).
(3.4)
|
∂xj
Now, we estimate
n ∂(ΦnV )j
∂(ΦnV )j ∂(Φn )j
∂(Φ
)
j
.
,
, and −
∂xi
∂xi
∂xi
∂xi When n = 1,
∂(ΦV )j ∂Φj (xV , x∗Vb ) ≤ kΦkΓ(i − j).
∂xi = ∂xi
∂Φ
The same estimation holds for | ∂xij |.
When n = 2,
∂(Φ2V )j ∂(ΦV )j (ΦV (xV )) X ∂(ΦV )j ∂(ΦV )k =
∂xi = ∂xi
∂xk
∂xi k∈V
X
≤
kΦk2 Γ(j − k)Γ(k − i) ≤ kΦk2 Γ(i − j)
k∈V
16
M. JIANG AND R. DE LA LLAVE
Inductively, we have
∂(ΦnV )j n
(3.5)
∂xi ≤ kΦk Γ(i − j).
The same estimation holds for
∂(Φn )j
.
∂xi
The difference
∂(ΦnV )j
∂(Φn )j
−
∂xi
∂xi
can also be estimated inductively.
When n = 1, we have
∂Φj (xV , x∗ ) ∂Φ (x̄) X ∂ 2 Φ
b
j
j
V
−
|
|d((xV , x∗Vb )k , xk )
≤
∂xi
∂xi ∂x
∂x
i
k
d
k∈Z
X
∂ 2 Φj
≤L
|
|≤L
kΦk min{Γ(i − j), Γ(k − j)}
∂xi ∂xk
b
b
k∈V
k∈V
X
≤ C1 exp(−α1 |i − j|)
exp(−β1 |i − k|)
X
k∈Vb
≤ C1 exp(−α1 |i − j|) exp(−β1 |i − ∂V |),
for some positive constants C1 , α1 , and β1 . The term |i − ∂V | denotes the minimal
distance between i and the boundary of V , ∂V .
Assume that this estimation holds for n − 1:
∂(Φn−1
∂(Φn−1 )j V )j
−
≤ Cn−1 exp(−αn−1 |i − j|) exp(−βn−1 |i − ∂V |),
∂xi
∂xi for some positive constants Cn−1 , αn−1 , and βn−1 depending on n, kΦk, and L.
n ∂(ΦnV )j
∂(Φ
)
∂
∂
j
n−1
∗
n−1
∂xi − ∂xi = | ∂xi Φj (ΦV (xV ), xVb ) − ∂xi Φj (Φ (x̄))|
X ∂Φj ∂(Φn−1
∂Φj ∂Φn−1
V )k
k
≤
−
∂xk (Φn−1 (xV ),x∗ ) ∂xi
n−1 (x̄)
∂x
∂x
Φ
k
i
V
b
V
k∈V
n−1 X ∂Φj ∂Φ
k
+
∂xk Φn−1 (x̄) ∂xi .
k∈Vb
LINEAR RESPONSE FUNCTION
17
We estimate these sums separately. We estimate the second sum first.
X ∂Φj X
∂Φn−1
k
kΦkΓ(k − j)kΦkn−1 Γ(i − k)
∂xk Φn−1 (x̄) ∂xi ≤
k∈Vb
k∈Vb
≤ kΦkn exp(−α|i − j|)
X
1
Γ 2 (i − k).
k∈Vb
For the first sum, we have
X ∂Φj ∂(Φn−1
∂Φj ∂Φn−1
V )k
k
−
n−1
∂xk (Φn−1 (xV ),x∗ ) ∂xi
∂x
∂x
Φ
(x̄)
k
i
V
b
V
k∈V
X ∂Φj
∂(Φn−1
∂Φ
)
j
k
V
(3.6)
n−1
n−1 (x̄) · ≤
|
−
|
∗
Φ
∂xk (ΦV (xV ),xVb ) ∂xk
∂xi k∈V
n−1 ∂Φj
∂(Φn−1
)
∂Φ
k
V
k
.
|Φn−1 (x̄) · −
+ ∂xk
∂xi
∂xi The second part of the sum (3.6) can be estimated as follows:
n−1 X ∂Φj ∂(Φn−1
)
∂Φ
k
V
k
−
∂xk Φn−1 (x̄) ∂xi
∂x
i
k∈V
X
≤
kΦkΓ(j − k)Cn−1 exp(−αn−1 |i − k|) exp(−βn−1 |i − ∂V |)
k∈V
≤ kΦkCn−1 exp(−βn−1 |i − ∂V |)
X
Γ(j − k) exp(−αn−1 |i − k|)
k∈V
≤ Cn exp(−βn−1 |i − ∂V |) exp(−αn |i − j|).
The first part of the sum (3.6) needs to be estimated more carefully.
X ∂Φj
∂(Φn−1
∂Φj
V )k |Φn−1 (x̄) ∗ ) −
(x
),x
∂xk |(Φn−1
V
V
b
V
∂xk
∂xi k∈V
X X ∂ 2 Φj n−1
∗
n−1
n−1
=
∂xk ∂xl d((ΦV (xV ), xVb )l , (Φ (x̄))l )kΦk Γ(i − k)
k∈V l∈Zd
X
XX
=
kΦk min{Γ(j − k), Γ(j − l)}C(n − 1)
Γ(l − j 0 )kΦkn−1 Γ(i − k)
k∈V l∈V
+
XX
j 0 ∈Vb
kΦk min{Γ(j − k), Γ(j − l)}LkΦkn−1 Γ(i − k).
k∈V l∈Vb
Both sums in the previous expression are bounded by
C exp(−α|i − j|) exp(−β|i − ∂V |),
18
M. JIANG AND R. DE LA LLAVE
for some constants C, α, and β depending on n, kΦk, and L. Thus, we have
n ∂(ΦnV )j
∂(Φ
)
j
≤ Cn exp(−αn |i − j|) exp(−βn |i − ∂V |),
(3.7) −
∂xi
∂xi for some positive constants Cn , αn , and βn depending on n, kΦk, and L. The statement
(3) follows the estimates (3.2), (3.3) (3.4) (3.5),and (3.7).
The last statement (4) of the lemma follows immediately from our previous estimates.
X
n
∂
∂ϕ ∂(ΦV )k n ∂xi ϕ(ΦV ) = ∂xk ∂xi k∈V
X
≤
kϕkΓ(k)kΦkn Γ(i − k) ≤ kϕkkΦkn Γ(i).
k∈V
Now, the only thing left to show is that for each i
s
lim XV,i
= Xis
V →Zd
uniformly in x̄. We will need the closeness of stable and unstable subspaces for ΦV,
and Φ in compact metric (2.1) ρq for some 0 < q < 1. Using the Grassmanian distance on subspaces of the manifold M , we introduce the corresponding ρq metric for
Hölder continuous sections of the tangent bundles close to ⊗i∈Zd Eis and ⊗i∈Zd Eiu . We
formulate these properties in the following lemma but omit their proofs as they are
proved using standard techniques (the fixed point theorem of contracting maps in the
compact metric ρq , see for example, Lemma 2 of [9]). We take V = Vm , the hypercubes
in Zd for convenience. EVs , EVu , E s , E u denote stable and unstable subspaces of ΦV and
Φ, respectively. The map ΦV is extended to the entire M by adding copies of f to it
outside the hypercube Vm .
Lemma 3.
(1) There exist constants C > 0, 0 < λ < 1 such that
ρq (hΦVn , hΦ ) ≤ Cλn .
(2) There exist constants C > 0, 0 < λ < 1 such that
ρq (EVsn , E s ) ≤ Cλn ,
ρq (EVun , E u ) ≤ Cλn .
(3) limVn →Zd XVsn ,i (xVn ) = Xis (x̄) uniformly in x̄.
LINEAR RESPONSE FUNCTION
19
Proof of the convergence. By Lemmas 2 and 3, we have
lim < grad(ϕV (ΦnV, )), XVs >= lim
V
→Zd
V
=
→Zd
X ∂ϕV (ΦnV, )
i∈V
∂xi
s
XV,i
X ∂ϕ(Φn )
Xis =< grad(ϕV (Φn )), X s >,
∂xi
d
i∈Z
i.e.,
Z
lim
V
∈Zd
<
grad(ϕV (ΦnV, )), XVs
Z
> dµΦV, =
< grad(ϕV (Φn )), X s > dµΦ .
3.4. Proof of Proposition 3: Convergence of the second term. To show the
convergence of the second series
∞ Z
∞ Z
X
X
u u
n
ϕ(ΦV, )divσ XV dµΦV, →
ϕ(Φn )divuσ X u dµΦ ,
n=0
n=0
the termwise convergence as V → Zd is proved exactly in the same way since we have
the divergence formula [14]
Z
Z
n
u
(3.8)
< gradϕ(ΦV, ), XV > dµΦV, = − ϕ(ΦnV, )divuσ XVu dµΦV, .
But the convergence of the series uniform in the volume V requires a careful analysis.
We need to estimate the term
Z
ϕ(ΦnV, )divuσ XVu dµΦV,
using the exponential decay of correlation functions for SRB measures µΦV, . The
meaning of the divergence divuσ XVu is given by Ruelle in [16] and we will also briefly
discuss it later.
We decompose the vector field in a different way. We project each component of XV
onto the unstable subspace first and then, add them up:
X
X
u
XVu = (
XV,i )u =
XV,i
,
i∈V
i∈V
20
M. JIANG AND R. DE LA LLAVE
u
i.e., XV,i
denotes the projection of the ith coordinate of XV onto the unstable subspace
u
EV . Thus,
Z
XZ
u u
n
u
ϕ(ΦV, )divσ XV dµΦV, =
ϕ(ΦnV, )divuσ XV,i
dµΦV, .
i∈V
Now it suffices to prove the following theorem.
Theorem 4. There exist constants C > 0, 0 < α, β < 1 that depend on the map Φ,
the vector field X, and the decay function Γ but are independent of V such that
Z
u
ϕ(ΦnV, )divuσ XV,i
≤ Ce−αn−β|i| .
dµ
Φ
V, This theorem follows from results of Bricmont and Kupiainen in [3, 4] once we show
u
that the function divuσ XV,i
(xV ) is Hölder continuous in xV with the Hölder constant
and exponent independent of the volume V with respect to the (translated) metric ρq,i :
ρq,i (xV , yV ) = sup q |i−j| d(xj , yj ).
j∈V
Since the vector field X is assumed to be translation invariant, we have
u
divuσ XV,i
= divuσ XVu −i,0 ,
where V − i denotes the volume V translated by −i. So, we just need to prove the
following theorem.
Theorem 5. There exist constant C > 0 and 0 < α < 1 independent of the volume V
such that
u
u
|divuσ XV,0
(xV ) − divuσ XV,0
(yV )| ≤ C[ρq (xV , yV )]α .
Now we focus on the proof of this theorem. We first need to recall the definition of
u
u
is usually not differentiable even though XV,0 , the coordinate
divuσ XV,0
. Note that XV,0
d
of XV at 0 ∈ Z is. However, the divergence, which measures the change of the
volume form under the flow determined by the vector field, exists for non-differentiable
vector fields satisfying certain conditions. We will see that since the stable foliation
is absolutely continuous, the divergence of any smooth vector field projected onto the
unstable manifold can be properly defined and calculated with the help of the holonomy
map defined by the stable foliation. Ruelle showed in [16] that using a specially adapted
u
can be expressed as the derivative of
local coordinate system, the divergence divuσ XV,0
the Jacobian of the holonomy map of the stable foliation. The choice of the adapted
coordinate system depends on the vector field XV,0 and is not convenient in our case
since we need to take the limit V → Zd . We use Ruelle’s idea of calculation of the
LINEAR RESPONSE FUNCTION
21
u
divergence to obtain a more general relation between the divergence divuσ XV,0
and the
Jacobian of the holonomy map. Since the divergence is independent of the choice of
the coordinate system, we identify a neighborhood on the Riemannian manifold with
a neighborhood of Rn = Ru ⊕ Rs equipped with Riemannian metric and the unstable
manifolds are identified with Ru .
Definition 6. Let X be a continuous vector field defined in an open set U on a Riemannian manifold with a volume form dw. Let G denote the family of C ∞ functions
in U with compact support. Assume that there is an integrable function h(x) such that
Z
Z
< gradg, X > dw = − g(x)h(x)dw,
for all g ∈ G. Then, we call h(x) the divergence of X and denote it by divX.
It is clear that the divergence of a vector field is unique up to a measure zero set.
When X is C 1 , the definition is consistent with the common definition if we choose
h(x) to be continuous. The following proposition can be directly verified and it gives
the geometric meaning of the divergence.
Proposition 7. Let X be a continuous vector field on Rn with a measure m determined
by a volume form dω. Let t ∈ (−1, 1) and At be a map on Rn defined by
At (x) = x + tX(x).
Assume that the induced measure A∗t m is absolutely continuous with respect to m with
a Radon-Nikodym derivative ρ(x, t) continuous in x. Assume that 1t (ρ(x, t) − ρ(x, 0)) =
1
(ρ(x, t) − 1) is bounded in x for all t sufficiently
small. Then, the divergence of X(x)
t
d
is equal to the derivative divX = dt ρ(x, t)t=0 when it exists.
When X is differentiable, the Radon-Nikodym derivative becomes the Jacobian of
At . Using the definition of the divergence, one can easily see that linearity holds, i.e.,
div(aX1 + bX2 ) = adivX1 + bdivX2 .
The Gauss divergence theorem can also be verified.
We prove the following lemma.
Lemma 4. Let X be a C 1 - vector field on Rn = Ru ⊕ Rs and E s (x) denote the distribution tangent to an absolutely continuous foliation W with smooth leaves transversal
to the manifold Ru . Let X = X u + X s denote the projections of X onto Ru and E s (x),
respectively. Then, divu X u , the divergence of X u on the manifold Ru with respect to
the volume form restricted to the manifold is well defined. Moreover, when the Jacobian
22
M. JIANG AND R. DE LA LLAVE
Jξ(xu , t) of the the holonomy map ξ(xu , t) from Ru to Ru + tX defined by the foliation
is differentiable in t and continuous in xu , we have
d
Jξ(xu , t)|t=0 , xu ∈ Ru
dt
u
where Xc is the projection of the vector field X onto the coordinate subspaces Ru and
Rs : X = Xcu + Xcs .
divu X u = divu Xcu −
Proof. Note that since the foliation W has differentiable leaves, the derivative dtd Jξ(xu , t)|t=0
is uniquely determined by its first order linear approximation, the distribution E s (xu ), xu ∈
Ru . Thus, in calculating the Jacobian of the holonomy map at t = 0, we may consider ξ(xu , t) as the intersection of hyperplanes E s (xu ) and Ru + tX. For each fixed
xu ∈ Ru , ξ(xu , t) is linear in t. Indeed, since any point in E s (xu ) can be expressed as
(xu + B(xu )xs , xs ) where B(xu ) is a linear map from Rs to Ru and Ru + tX = Ru + tXcs ,
we have, in the coordinate system of Ru × Rs ,
ξ(xu , t) = (xu + tB(xu )Xcs , Xcs ).
Thus, the holonomy map maps xu to xu + tB(xu )Xcs , and by Proposition 7,
d
Jξ(xu , t)|t=0 = div B(xu )Xcs .
dt
We can now determine X u in terms of Xcu , Xcs and B(xu ). We look for coefficients
a and b such that X = aeu + bf s , where eu , f s denotes the basis of Ru and E s (xu ):
f s = (B(xu )es , es ). We have equations:
Xcu = a + bB(xu ); Xcs = b.
Thus,
X u = a = Xcu − B(xu )Xcs .
Note that Xcu is C 1 . We have
divX u = div(Xcu − B(xu )Xcs ) = divXcu −
d
Jξ(xu , t)|t=0 .
dt
Remark. We have assumed in the proof that the volume forms on Ru and Ru + tX
are the same. This is not true in general. But this will not affect the formula as we
are taking limit t → 0. Note also that divXcu is not zero in general, even when Xcu is
constant since the volume form on Ru is not necessarily constant. But it is possible to
choose a coordinate system such that divXcu = 0 locally.
LINEAR RESPONSE FUNCTION
23
Proof. We now prove Theorem 5.
u
In the context of our coupled hyperbolic attractors, we need to show that divuσ XV,0
depends on xV Hölder continuously in the metric ρq , where σ is the canonical metric
on the unstable manifold. We note that the density function sV of this metric satisfies
the equation
∞
i−1
sV (xV ) Y h J u ΦV (Φ−k
V xV )
=
.
sV (yV ) k=1 J u ΦV (Φ−k
y
)
V
V
We also have the identity
u
u
u
divuσ XV,0
= divuω XV,0
+ < grad log sV , XV,0
>,
where w is the induced volume form on the unstable manifold. We will drop w for
simplicity.
Note that the gradient
grad log sV = grad log
∞
Y
J u ΦV (Φ−k
V xV )
k=1
u
is taken in the unstable direction XV,0
. We can prove its Hölder continuity with straightu
forward estimations as we did in Part I. For divu XV,0
, we use the representation from
Lemma 4
d
u
u
− JξV (xV , t),
divu XV,0
= divu XV,0,c
dt
where ξV (xV , t) denotes the corresponding holonomy map defined on the unstable manu
ifold of ΦV . Note that the first term XV,0,c
is differentiable in xV and its partial derivatives with respect to xi approach zero exponentially as |i| → ∞. This guarantees the
Hölder continuity of this divergence and the Hölder constant and exponent can be chosen independent of V . For the second term, we need an explicit representation of the
Jacobian.
1. Calculating the derivative of the Jacobian with respect to t.
We identify a small neighborhood of the manifold with Ru × Rs . We shall now
consider the Jacobian whose formula is given [11, p. 190] by
∞
|
det(D
Φ
pn V )|
Y
En ,
JξV (xV , t) =
n=0 | det(Dp0n ΦV )|
En0
u
where pn = ΦnV (xV ), p0n = ΦnV (ξ(xV , t)), En = Dpn ΦnV EVu (xV ), and En0 = Dp0n ΦnV EV,t
,
u
u
where EV,t = EV (xV ) + tXV,0 (xV ).
24
M. JIANG AND R. DE LA LLAVE
It is clear that this Jacobian is bounded away from zero since it is equal to 1 when
t = 0. We prove the Hölder continuity of
d
log JξV (xV , t)
dt
t=0
in terms of xV with respect to the metric ρq with Hölder exponent and constant independent of V .
Note that the numerator det(Dpn ΦV ) is independent of t. So, we have
En
∞
X
d
d
log JV (x, t) = −
log | det(Dp0n ΦV )|.
dt
dt
t=0
En0
n=0
We now consider dtd log det(Dp0n ΦV ) with n and xV fixed. Note that when t varies,
0
(3.9)
En
u
(x̄)
p0n = ΦnV (ξ(x̄, t)) is moving along the stable manifold. The subspace Dp0n ΦV EV,t
varies smoothly in t.
We denote ψ(y, E) = log | det(Dy ΦV )|, where Dy ΦV is the differential of ΦV
E
E
along the subspace E at the point y in a neighborhood U of pn , and E is any subspace in
a CΓα -neighborhood V (using the Grassmannian distance) of the direct sum ⊗i∈V Eiu (y).
u
We take V large enough to include all subspaces EV,t
for 0 ≤ t ≤ t0 (for some t0 > 0).
Thus, we have a smooth function ψ(y, E) on the product space U × V with differentiable manifold structure.
Note that
u
s
= tXV,0 (xV ) + EVu (xV ) = tXV,0
(xV ) + EVu (xV )
EV,t
and that
u
s
Dp0n ΦnV EV,t
= tDp0n ΦnV XV,0
(xV ) + Dp0n ΦnV EVu (x̄).
Taking the derivative of log | det(Dp0n ΦV )| with respect to t, we have
En0
d
∂
d
∂
s
0
| det(Dpn ΦV )| =
ψ(y, E)DΦnV ξ(x̄, t)+
ψ(y, E)Dp0n ΦnV XV,0
(x̄),
0
dt
∂y
dt
∂E
En
∂
∂
where the differentials ∂y
ψ(y, E) and ∂E
ψ(y, E) are partial differential operators of
ψ(y, E) taken with respect to y and E, respectively.
At the point t = 0, since the stable subspace EVs is tangent to the leaf of the stable
foliation passing the point x̄, we simply have
d
s
ξV (xV , 0) = XV,0
dt
LINEAR RESPONSE FUNCTION
25
and therefore,
∂
d
∂
s
ψ(y, E)DΦnV ξV (xV , t) =
ψ(y, E)DΦnV XV,0
.
∂y
dt
∂y
Evaluating the derivative at t = 0, we have
∂
∂
d
s
s
log | det(Dp0n ΦV )| =
ψ(y, E)
DΦnV XV,0
(x̄)+
ψ(y, E) DΦnV XV,0
(x̄).
dt
∂y
∂E
En0
t=0
y=pn
En
2.Proof of the Hölder continuity of the derivative
We need a standard lemma.
P
Lemma 5. Let ∞
n=1 φn (x) be a series of Hölder continuous functions with a common
Hölder exponent α and Hölder constants cn satisfying the condition cn < Ln for some
constant L > 0. Assume that |ψn (x)| < Cλn for another constants C > 0, 0 < λ < 1.
Then, the sum is a Hölder continuous with Hölder exponent and constant dependent
on constants α, L, C and λ.
s
Since XV,0
is in the stable subspace and Φ ∈ CΓr (M, M), we have
s
s
kDΦnV XV,i
(xV )k ≤ Cλn kXV,0
k
for some constants C > 0, 0 < λ < 1.
We now need to consider partial derivatives of the function
ψ(y, E) = log | det(Dy ΦV )|.
E
Using the basis of the tangent space, we represent the differential Dy ΦV as a
E
matrix.
Note that y is changing along a stable manifold of ΦV . So, we can use the coordinate system provided by the stable and unstable subspaces ⊕i∈V Eis , ⊕i∈V Eiu of the
u
uncoupled map FV at the corresponding conjugating point h−1
V y. Let {ei , i ∈ V } and
{esi , i ∈ V } denote the orthonormal bases of the direct sums ⊕i∈V Eiu and ⊕i∈V Eis .
We take an orthonormal basis of E, {ẽui }. Since E is in the CΓα - neighborhood V of
⊕i∈V Eiu , {ẽui } can be uniquely represented as a linear combination of {eui , i ∈ V } and
{esi , i ∈ V }:
X
(3.10)
ẽui =
aij eui + bij esi ,
j∈V
where aij and bij satisfy the conditions
kaij − δij k ≤ CΓ(i − j), kbij k ≤ CΓ(i − j).
26
M. JIANG AND R. DE LA LLAVE
The constants C and are valid for the entire neighborhood of ⊕i∈V Eiu .
The matrix representation of Dy ΦV |E is now given by the images of the basis vectors
under the map DΦV , i.e.,
(DΦV ẽui ), i ∈ V.
If we use the same letter E to denote the coefficient matrix in (3.10), then, the matrix
representation is simply
DΦV E.
If the unstable subspace Eiu is of dimension u and the manifold M is of dimension n,
then DΦV E is a n|V | × u|V | matrix. We now have
[det(Dy ΦV |E )]2 = det(DΦV E)T DΦV E = det E T (DΦV )T DΦV E,
where DΦV is evaluated at the point y.
This determinant as a function of the point y is well defined in an open neighborhood
of the hyperbolic attractor ∆F . For each point y, it is a function of E in the CΓα neighborhood V of the unstable subspace at h−1
V y. We prove the following estimates
on its partial derivatives. These estimates allow us to verify the conditions of Lemma
5 and thus, obtain the desired Hölder continuity.
Lemma 6. Assume that the map Φ ∈ CΓr (M, M), r > 4. Then, there exist constants
C and β independent of V such that
(3.11)
k
∂ 2 ψ(y, E)
k ≤ Ce−β|k−j| , k, j ∈ V
∂yk ∂yj
and
(3.12)
k
∂ 2 ψ(y, E)
k ≤ Ce−β|k| , k ∈ V.
∂yk ∂E
Proof. We rewrite the matrix B T B in the form
B T B = E T (DΦV )T DΦV E = (DF )(I + A),
where (DF ) is a diagonal matrix dependent on FV and E, but independent of the
perturbation.
By the Banach algebra property of the decay function Γ, we know that the offdiagonal entries of the matrix A satisfies the same decay property as those of DΦV and
E.
We now calculate the partial derivatives of log det(DF )(I + A).
log det(DF )(I + A) = log det(DF ) + log det(I + A).
LINEAR RESPONSE FUNCTION
27
Clearly, when k 6= j,
∂ 2 log det(DF )
= 0.
∂xk ∂xj
We expand the function log det(I + A):
log det(I + AV ) = trace(log(I + AV )) = −
X
wi (xV ),
i∈V
where
wi (xV ) =
∞
X
(−1)n
n=1
n
(n)
aii (xV ),
(n)
where {aii (xV ), i ∈ V } are the entries on the main diagonal of the matrix (AV )n .
It suffices now to show that there exist constants C, β1 , β2 such that
k
(3.13)
∂ 2 wi
k ≤ Ce−(β1 |i−k|+β2 |i−l|) .
∂xk ∂xl
(n)
We denote the entries of (AV )n by aij (xV ). The following lemma is similar to what
were proved in earlier papers [7, 9] about the same matrix and the estimation (3.13)
follows immediately.
Sublemma 1. The entries of the matrix (AV )n satisfy the following properties.
(n)
(1) |aij (xV )| ≤ (c)n C1 exp(−β1 |i − j|).
(n)
∂aij
(xV )| ≤ (c)n C2 exp(−β2 (|i − j| + |j − k| + |i − k|)).
∂xk
(n)
∂ 2 aij
| ∂xk ∂x
(xV )| ≤ (c)n C3 exp(−β3 (|i − j| + |i − k| + |i − l| + |j
l
(2) |
(3)
− k| + |j − l| + |k − l|)).
We leave out the proof since it involves only straightforward estimations.
We are left to calculate the partial derivative of
1
log det E T (DΦV )T DΦV E
2
with respect to E. For convenience, we denote the matrix (DΦV )T DΦV by B again.
We have:
ψ(ȳ, E) =
log det(E + tE 0 )T B(E + tE 0 ) = log det(E T BE + t(E 0T BE + E T BE 0 ) + t2 E 0T BE 0 )
= log det E T BE
+ log det(I + t(E 0T BE(E T BE)−1 + (E T BE)−1 E T BE 0 ) + t2 E 0T BE 0 (E T BE)−1 )
Thus, the derivative of 2ψ(ȳ, E) along the direction E 0 is equal to
trace E 0T BE(E T BE)−1 + (E T BE)−1 E T BE 0 .
28
M. JIANG AND R. DE LA LLAVE
2
With E, E 0 fixed, ∂ ∂yψ(y,E)
satisfies the estimate (3.12) in the lemma as claimed since
k ∂E
∂B
satisfies the same estimate and is independent of the volume V .
∂yk
u
(xV )
By Lemmas 5 and 6, we conclude that the Jacobian dtd JξV (xV , t) and thus,divu XV,0
t=0
is a Hölder continuous function with respect to a compact metric ρq with the Hölder
constant and exponent independent of the volume V . This completes our proof of the
formula for the linear response function.
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Department of Mathematics, Wake Forest University, Winston Salem, NC 27109,
USA
E-mail address, M. Jiang: [email protected]
Department of Mathematics, University of Texas, Austin, TX 78712-0257, USA
E-mail address, R. de la Llave: [email protected]