\magnification=1200
\def\qed{\unskip\kern 6pt\penalty 500\raise -2pt\hbox
{\vrule\vbox to 10pt{\hrule width 4pt\vfill\hrule}\vrule}}
\rightline{\it Dedicated to Klaus Hepp}
\rightline{\it and Walter Hunziker}
\rightline{\it for their 60-th birthday.}
\null\vskip 3truecm
\centerline{DIFFERENTIATION OF SRB STATES.}
\bigskip
\bigskip
\centerline{by David Ruelle\footnote{*}{IHES (91440 Bures sur Yvette,
France) $<$ruelle@ihes.fr$>$, and Math. Dept., Rutgers University
(New Brunswick, NJ 08903, USA).}}
\bigskip
\bigskip
\indent
{\it Abstract.} Let $f$ be a diffeomorphism of a manifold $M$,
and $\rho_f$ a (generalized) SRB state for $f$. If ${\rm supp}\rho_f$
is a hyperbolic compact set we show that the map $f\mapsto\rho_f$ is
differentiable in a suitable functional setup, and we compute the
derivative. When ${\rm supp}\rho_f$ is an attractor, the derivative is
given by
$$ \delta\rho_f(\Phi)=\sum_{n=0}^\infty\rho_f
\langle{\rm grad}(\Phi\circ f^n),X\rangle $$
where $X$ is the vector field $\delta f\circ f^{-1}$. This formula
can be extended, at least formally, to time dependent situations, and
also to nonuniformly hyperbolic situations.
\medskip
The above results will find their use in the study of the
Onsager reciprocity relations and the fluctuation-dissipation formula of
nonequilibrium statistical mechanics.
\vfill\eject
\noindent
{\bf 0. Introduction.}
\bigskip
\bigskip
In a recent paper [7], G.Gallavotti has outlined a new proof of
Onsager's reciprocity relations, based on the study of the SRB measure
$\rho_f$ for a hyperbolic dynamical system $(M,f)$. To give a rigorous
and general version of Gallavotti's argument, one has to study the
dependence $f\mapsto\rho_f$, and in particular compute the derivative.
In fact, one may argue that these problems are at the core of
nonequilibrium statistical mechanics; they are the subject of the
present paper. We do not make here the assumption of [7] that we are
close to a Hamiltonian situation (where $f$ has a smooth invariant
measure); our analysis will thus be valid "far from equilibrium". In
what follows we concentrate on the mathematics, and leave the application
to nonequilibrium statistical mechanics for other occasions.
\medskip
Let $K$ be a mixing Axiom A attractor for the diffeomorphism
$f$. In a suitable functional setup we shall show that the SRB state
$\rho_f$ on $K$ depends differentiably on $f$. A variation $\delta f$
of $f$ corresponds to a vector field $X=\delta f\circ f^{-1}$, and we
shall obtain the formula
$$ \delta\rho_f(\Phi)
=\sum_{k=0}^\infty\rho_f(\langle\hbox{grad}(\Phi\circ f^k),X\rangle) $$
This formula is relatively easy to guess, but its proof requires some
care. Instead of the Axiom A attractor case we shall in fact deal with
the more general situation where $K$ is a hyperbolic set with local
product structure, and $\rho_f$ the corresponding generalized SRB state
(Sections 1, 2 and 3). In Section 4 we shall see how the definition of
attractor and of SRB state can be extended to a general bounded
time dependent perturbations of $f$. Finally, in Section 5 we shall
discuss a formula for the formal derivative of the SRB state $\rho_f$
with respect to $f$, without uniform hyperbolicity assumption.
\medskip
The rest of this introduction is a brief summary of facts
concerning hyperbolic sets. For more details see Smale [20], Shub [16],
Ruelle [14], and references quoted there.
\medskip
{\sl Hyperbolicity.}
\medskip
Let $K$ be a compact invariant set for the diffeomorphism $f$ of
a finite-dimensional manifold $M$, we assume $f$ to be of class
$C^r$, with $r\ge 1$. We choose some Riemann metric on $M$.
Suppose that $T_K M$ (the tangent bundle restricted to $K$) has a
continuous $Tf$-invariant splitting $T_K M=V^-\oplus V^+$ and that there are
constants $C\ge1$, $\theta>1$ such that
$$ \max_{x\in K}\|(T_xf^{\mp n}|V^{\pm}(x))\|\le C\theta^{-n} \qquad
\hbox{for} \quad n\ge0 $$
Then $K$ is called a hyperbolic (compact invariant) set for $f$. We
call $V^-=V^s$ and $V^+=V^u$ the stable and unstable subbundles
respectively.
\medskip
Local stable manifolds ${\cal V}^-(x)={\cal V}^s(x)$ and unstable
manifolds ${\cal V}^+(x)={\cal V}^u(x)$ are defined by
$$ {\cal V}^\pm(x) =\{y\in M:\,d(f^{\mp n}y,f^{\mp n}x)1$ such that if $y,z\in{\cal
V}^\pm(x)$,
$$ d(f^{\mp n}y,f^{\mp n}z)\le C'\theta'^{-n}d(y,z)\qquad
\hbox{for}\quad n\ge0 $$
\medskip
{\sl Expansiveness, H\"older continuity of hyperbolic splitting,
Axiom A attractors.}
\medskip
The map $f$ restricted to the hyperbolic invariant set $K$ is an
{\it expansive homeomorphism}. This means that $d(f^kx,f^ky)<\epsilon$
for all $k\in{\bf Z}$, implies $x=y$.
\medskip
If $r>1$, the stable and unstable subbundles $V^\pm$ are H\"older
continuous, {\it i.e.}, the sections $x\mapsto V^\pm(x)$ of the Grassmannian
over $K$ are $C^\alpha$ for some $\alpha>0$.
\medskip
We say that the compact hyperbolic $f$-invariant set $K$ is
{\it transitive} if $K$ contains a dense orbit $(f^ka)_{k\in{\bf Z}}$.
We say that $K$ is an {\it Axiom A attractor} if $K$ is transitive and
has an open neighborhood $U$ such that
$$ \cap_{n\ge0}f^nU=K $$
It follows that the local unstable manifolds ${\cal V}^u(x)$ of points of $K$
lie in $K$ (this is also true for the global unstable manifolds
$\cup_{n=1}^\infty f^n{\cal V}^u(x)$). One can then show that the
$f$-periodic points are dense in $K$. The local stable
manifolds ${\cal V}^s(x)$ of points of $K$ fill a neighborhood (say $U$)
of $K$. Consider a continuous map $\phi:S_1\to S_2$ along the
${\cal V}_x^s$ between two smooth transverse sections $S_1$
and $S_2$ (for instance two pieces of unstable manifolds). One can show
that $\phi$ is H\"older continuous, and absolutely continuous (for the
Riemann volume elements of $S_1$, $S_2$) with H\"older continuous Jacobian.
\medskip
{\sl Local product structure, shadowing.}
\medskip
We say that the compact hyperbolic $f$-invariant set $K$ has
local product structure if $R$ can be chosen in the definition of ${\cal
V}^\pm(x)$ such that, for all $x,y\in K$
$$ {\cal V}^-(x)\cap{\cal V}^+(y)\subset K $$
In particular, an Axiom A attractor has local product structure. For small
$R$, we may assume that the ${\cal V}^\pm(x)$ are nearly flat, so that
${\cal V}^-(x)\cap{\cal V}^+(y)$ consists of at most one point. One
can check that the map $(x,y)\mapsto[x,y]$, where $[x,y]$ is the only
point in ${\cal V}^-(x)\cap{\cal V}^+(y)$, defines a product structure
in a neighborhood of each point of $K$.
\medskip
A remarkable feature of hyperbolic sets with local product
structure is that $\delta$-pseudo- orbits are well approximated by true
orbits. We say that $(x_k)_{k\in[k_0,k_1]}$ is a $\delta$-{\it
pseudoorbit} for $f$ if $d(fx_k,x_{k+1})<\delta$ for every finite
$k\in[k_0,k_1-1]$, where $k_0$, $k_1$ may be finite or $\pm\infty$. The
pseudoorbit $(x_k)$ is $\epsilon$-shadowed by the orbit $(f^kx)$ if
$d(f^kx,x_k)<\epsilon$ for all $k\in[k_0,k_1]$. Bowen has proved the following
{\it shadowing lemma:
\medskip
Let $K$ be a hyperbolic set with local product structure for
$f$. For every $\epsilon>0$ there is $\delta>0$ such that every
$\delta$-pseudoorbit in $K$ is $\epsilon$-shadowed by an orbit in $K$.}
\medskip
This is a very efficient tool in the study of hyperbolic
systems; it was for instance used by Bowen [3] to prove the existence of
Markov partitions (first introduced by Sinai [17], [18]) in general and natural
fashion. For a discussion of Markov partitions and symbolic dynamics we
must however refer to the original papers.
\vfill\eject
\noindent
{\bf 1. Stuctural stability results.}
\bigskip
\bigskip
{\sl The spaces} ${\cal M, B, A}$.
\medskip
From now on we take $r$ integer$>1$, and let $K_0$ be
a hyperbolic set for $f_0$ of class $C^r$. Then, the stable and
unstable subbundles $V_0^\pm$ are $C^{\alpha}$ for some $\alpha>0$. The
$C^\alpha$ maps $K_0\to M$ form a Banach manifold ${\cal M}$. The maps
close to the inclusion map $K\hookrightarrow M$ are described by a chart of
${\cal M}$ which we may take to be the open $\epsilon$-ball $B$ around 0 in
a Banach space ${\cal B}$. Using the exponential map $TM\to M$, we
may take for ${\cal B}$ the space of $C^\alpha$ sections of $T_{K_0}M$.
Finally, we shall denote by ${\cal A}$ the space of $C^r$ diffeomorphisms
sufficiently close to $f_0$ in a fixed neighborhood $U$ of $K_0$ in $M$.
\medskip
{\sl 1.1 Proposition.}
\medskip{\it
Let $r\ge2$.
\medskip
(a) The map ${\cal A}\times{\cal M}\to{\cal M}$ defined by
$(f,j)\mapsto f\circ j\circ f_0^{-1}$ is $C^{r-1}$.
\medskip
(b) The tangent map ${\cal T}$ to $j\mapsto f\circ j\circ
f_0^{-1}$ is given by
$$ ({\cal T}_j\delta)(x)=(T_{j(f_0^{-1}x)}f)\delta(f_0^{-1}x) $$
where $\delta\in T_j{\cal M}$.}
\medskip
To prove (a), it will suffice to show that $(f,j)\mapsto f\circ
j$ is $C^{r-1}$. Furthermore the problem is local, {\it i.e.}, it
suffices to consider $j$ and $f\circ j$ near $x_0\in K_0$. The map
$f\mapsto f\circ j$ is $C^\omega$ (in fact linear, using suitable local charts).
Differentiating $k$ times $f\circ j$ with respect to $j$ introduces the
$k$-th derivative of $f$, which is $C^{r-k}$, and composed with $j$
this gives a $C^\alpha$ function if $r-k\ge1$. Therefore
$(f,j)\mapsto f\circ j$ is $C^{r-1}$ as announced.
\medskip
(b) follows directly from the definitions.\qed
\medskip
For the next proposition, remember that ${\cal A}$ is a {\it
sufficiently small} neighborhood of $f_0$.
\medskip
{\sl 1.2 Proposition.}
\medskip{\it
Let $r\ge2$.
\medskip
(a) The inclusion map $K_0\hookrightarrow M$ is a hyperbolic
fixed point of the map ${\cal M}\to{\cal M}$ defined by $j\mapsto
f_0\circ j\circ f_0^{-1}$.
\medskip
(b) For $f\in{\cal A}$, the map ${\cal M}\to{\cal M}$ defined by
$j\mapsto f\circ j\circ f_0^{-1}$ has a unique fixed point $j(f)$ close
to $K_0\hookrightarrow M$. This fixed point is hyperbolic and is a
$C^\alpha$ homeomorphism $K_0\to K=j(f)K_0$.
\medskip
(c) The map $f\mapsto j(f)$ is $C^{r-1}:\,{\cal A}\to{\cal M}$,
and the tangent map $\delta f\mapsto\delta j$ is given by
$$ \delta j=(1-{\cal T}_{j(f)})^{-1}(\delta f\circ f^{-1}\circ j(f)) $$}
\medskip
Clearly $K_0\hookrightarrow M$ is a fixed point of $j\mapsto
f_0\circ j\circ f_0^{-1}$. The corresponding tangent map is ${\cal
T}_0:\,{\cal B}\to{\cal B}$ given by
$$ ({\cal T}_0\delta)(x)=(T_{f_0^{-1}x}f_0)\delta(f_0^{-1}x) $$
(see Proposition 1.(b)). We have to show that this is a hyperbolic
linear map, {\it viz.}, its spectrum is disjoint from the unit circle.
Here we use the fact that the splitting of $T_{K_0}M$ into stable and
unstable subbundles is $C^\alpha$, giving a decomposition ${\cal B}={\cal
B}^s\oplus{\cal B}^u$ such that ${\cal T}_0|{\cal B}^s$ and ${\cal
T}_0^{-1}|{\cal B}^u$ have spectral radius $<1$. This proves (a).
\medskip
Using Proposition 1(a), Proposition 2(a), and the implicit
function theorem, we see that $j\mapsto f\circ j\circ f_0^{-1}$ has a
unique fixed point $j(f)$ close to $K_0\hookrightarrow M$. By
continuity, this fixed point is hyperbolic ({\it i.e.}, ${\cal
T}_{j(f)}$ is a hyperbolic linear map). By expansiveness of $f_0$ on
$K_0$, $j(f)$ cannot collapse different orbits, and is thus injective.
This proves (b).
\medskip
[We have here followed Hirsch and Pugh [8] in establishing the
persistence of the hyperbolic set $K$].
\medskip
The implicit function theorem also yields that $f\mapsto j(f)$
is $C^{r-1}$, and by differentiating $j\circ f_0=f\circ j$ we get
$$ \delta j\circ f_0=\delta f\circ j+Tf\circ\delta j $$
hence
$$ (1-{\cal T}_{j(f)})\delta j=\delta f\circ j\circ f_0^{-1}
=\delta f\circ f^{-1}\circ j $$
hence
$$ \delta j=(1-{\cal T}_{j(f)})^{-1}(\delta f\circ f^{-1}\circ j(f)) $$
proving (c).\qed
\medskip
{\sl 1.3 Proposition.}
\medskip{\it
Let $r\ge3$. We denote by $\pi:\widetilde M\to M$ the Grassmannian
of $TM$, and let $\tilde f:\widetilde M\to\widetilde M$ be induced by $Tf$.
Also let $\widetilde{\cal M}$ denote the Banach manifold of $C^\beta$ maps:
$K_0\to\widetilde M$, for some suitably small $\beta>0$ (we take
$\beta\le\alpha$).
\medskip
(a) The map ${\cal A}\times\widetilde{\cal M}\to\widetilde{\cal M}$
defined by $(f,\tilde\jmath)\mapsto\tilde f\circ\tilde\jmath\circ f_0^{-1}$
is $C^{r-2}$.
\medskip
(b) The canonical lifting $K_0\to V_0^u$ is a hyperbolic fixed
point of the map $\widetilde{\cal M}\to\widetilde{\cal M}$ defined by
$\tilde\jmath\mapsto\tilde f_0\circ\tilde\jmath\circ f_0^{-1}$.
\medskip
(c) For $f\in{\cal A}$, the map $\widetilde{\cal M}\to\widetilde{\cal
M}$ defined by $\tilde\jmath\mapsto\tilde f\circ\tilde\jmath\circ
f_0^{-1}$ has a unique fixed point $\tilde\jmath(f)$ close to $K_0\to
V_0^u$. Furthermore $\pi\circ\tilde\jmath(f)=j(f)$,
$\tilde\jmath(f)x=V^u(j(f)x)$, and $f\mapsto\tilde\jmath(f)$ is
$C^{r-2}$: ${\cal A}\to\widetilde{\cal M}$.}
\medskip
(a) is proved like Proposition 1.1(a), taking into account the
fact that $\tilde f$ is of class $C^{r-1}$.
\medskip
From the hyperbolic splitting $T_{K_0}M=V_0^s\oplus V_0^u$ (for
$Tf$), one also obtains a hyperbolic splitting $T_{V_0^u}\widetilde M
=\widetilde V_0^s\oplus\widetilde V_0^u$ (for $T\tilde f$). In fact
$$ \widetilde V_0^s=(T\pi|T_{V_0^u}\widetilde M)^{-1}V_0^s $$
and
$$ \widetilde{\cal V}_0^u=\{\xi:\pi\xi\in{\cal V}_0^u\hbox{ and
}\xi\hbox{ is the tangent space to }{\cal V}_0^u\hbox{ at }\pi\xi\} $$
Note that $x\mapsto\widetilde{\cal V}_0^u(x)$ is continuous because
$x\mapsto{\cal V}_0^u(x)$ is continuous $K\to C^r$. Therefore, the
splitting $\widetilde V_0^s\oplus\widetilde V_0^u$ is again
$C^\beta$ for some $\beta>0$, and (b) follows.
\medskip
Using (a), (b), and the implicit function theorem, we see that
$\tilde\jmath\mapsto\tilde f\circ\tilde\jmath\circ f_0^{-1}$ has a
unique fixed point $\tilde\jmath(f)$ close to $K_0\to V_0^u$. Since
$\pi\circ\tilde f=f\circ\pi$, we have
$$ \pi\circ\tilde\jmath(f)=\pi\circ\tilde f\circ\tilde\jmath(f)\circ f_0^{-1}
=f\circ\pi\circ\tilde\jmath(f)\circ f_0^{-1} $$
which shows that $\pi\circ\tilde\jmath(f)=j(f)$. Since $\widetilde
K=\tilde\jmath(f)K_0$ is $\tilde f$-invariant and close to $V_0^u$, we
have $\widetilde K=V^u$, {\it i.e.}, $\tilde\jmath(f)x=V^u(j(f)x)$.
Finally, the implicit function theorem also shows that
$f\mapsto\tilde\jmath(f)$ is $C^{r-2}$: ${\cal A}\to\widetilde{\cal M}$,
concluding the proof of (c).\qed
\vfill\eject
\noindent
{\bf 2. Generalized SRB measures: smooth dependence on $f$.}
\bigskip
\bigskip
We assume from now on that $K_0$ has local product structure,
and that $f_0|K_0$ is mixing (for instance $f_0$ satisfies Smale's Axiom
A, and $K_0$ is a mixing basic set). Then also $K=K_f=j(f)K_0$ has
local product stucture for $f$, and $f|K$ is mixing.
\medskip
If $f\in{\cal A}$, the (generalized) SRB measure\footnote{*}{SRB
mesures were introduced by Sinai [19] for Anosov diffeomorphisms and
extended to Axiom A attractors for diffeomorphisms (Ruelle [12]) and flows
(Bowen and Ruelle [5]). For the general situation where uniform
hyperbolicity is not required see Ledrappier and Young [10]. In this
Section and the next we consider another generalization where we assume
uniform hyperbolicity, but not attractivity. The uniqueness of $\rho$
maximizing (1) is because $\log J_f^u$ is H\"older continuous, and $f|K$
mixing (see Bowen [4], or Ruelle [13]).} with respect to $f$ on $K$ is
the unique equilibrium state for $-\log J_f^u$, {\it i.e.}, the unique
$f$-invariant probability measure $\rho=\rho_f$ on $K$ making
$$ h_f(\rho)-\rho(\log J_f^u) \eqno(1) $$
maximum. Here $h_f(\rho)$ is the {\it entropy} of $\rho$, and $J_f^u$
is the {\it unstable Jacobian} [therefore, $\rho(\log J_f^u)$ is the sum
of the positive Lyapunov exponents for $\rho$]. We do not make the
usual assumption that $K$ is an attractor\footnote{**}{When $K$ is not
an attractor, $\rho_f$ serves to describe diffusion away from $K$ under
$f$. This is the content of Proposition 3.1 in Ruelle [15]. See also
Bowen and Ruelle [5], Young [21], Lopes and Markarian [11] (for a
special case: open billiard described by a Cantor set), Eckmann and
Ruelle [6] Section IV E. The work by Kaplan, Yorke, Kantz, Grassberger,
Gaspard, and Nicolis should also be mentioned here.}. The maximum of (1)
is $P(\log J_f^u)\le0$ [the value $0$ is obtained if and only if
$K$ is an attractor, see [5]].
\medskip
Let $\bar\jmath(f):K\to K_0$ be the inverse of $j(f)$ considered
as a map $K_0\to K$, and define $\mu_f=\bar\jmath(f)^*\rho_f$. Then,
$\mu_f$ is the unique equilibrium state with respect to $f_0$ on $K_0$
for $-\log J_f^u\circ j(f)$. [This follows from $j(f)\circ f_0=f\circ
j(f)$].
\medskip
{\sl 2.1 Proposition.}
\medskip{\it
Let $r\ge3$. We assume that $K$ has local product structure
with respect to $f$, and that $f|K$ is mixing.
\medskip
(a) The map $f\mapsto J_f^u\circ j(f)$ is $C^{r-2}$:
${\cal A}\to C^\beta(K_0)$.
\medskip
(b) The map $f\mapsto\mu_f|C^\beta(K_0)$ is $C^{r-2}$:
${\cal A}\to C^\beta(K_0)^*$.}
\medskip
Let $u$ be the dimension of the unstable subspaces. We note
that $J_f^u\circ j(f)$ is the norm of $(Tf)^{\wedge u}$ evaluated at
$\tilde\jmath(f)$, and that $f\mapsto Tf$ is $C^\omega$: ${\cal A}\to
C^{r-1}$. Since, by Proposition 1.3(c), $f\mapsto\tilde\jmath(f)$ is
$C^{r-2}$: ${\cal A}\to\widetilde{\cal M}$, we see that
$f\mapsto J_f^u\circ j(f)$ is $C^{r-2}$: ${\cal A}\to C^\beta(K_0)$,
proving (a).
\medskip
We shall now use the fact that, if $I$ is the set of $f_0$-invariant
probability measures on $K_0$, then the {\it pressure}
$$ A\mapsto P(A)=\max_{\mu\in I}\,[h_{f_0}(\mu)+\mu(A)] $$
is a $C^\omega$ function on $C^\beta(K_0)$. Furthermore, the derivative
of $P$ at $A$ (which is an element of the dual $C^\beta(K_0)^*$) is the
restriction to $C^\beta(K_0)$ of the equilibrium state $\mu^A$ for $A$.
[For these results, see [13]]. Therefore the map
$A\mapsto\mu^A|C^\beta(K_0)$ is $C^\omega$: $C^\beta(K_0)\to
C^\beta(K_0)^*$. Applying this to $A=-\log J_f^u\circ j(f)$, and
$\mu^A=\mu_f$, we see (using (a)) that $f\mapsto\mu_f|C^\beta(K_0)$ is
$C^{r-2}$: ${\cal A}\to C^\beta(K_0)^*$, proving (b).\qed
\medskip
{\sl 2.2 Proposition.}
\medskip{\it
Let $r\ge3$. The map $f\mapsto\rho_f|C^{r-1}(M)$ (where
$\rho_f$ is the SRB state for $f$) is $C^{r-2}$: ${\cal A}\to C^{r-1}(M)^*$.}
\medskip
We use the fact that $\rho_f=j(f)^*\mu_f$, so that
$$ \rho_f|C^{r-1}(M)=\ell(f)^*(\mu_f|C^\beta(K_0)) $$
where the bounded operator $\ell(f):C^{r-1}(M)\to C^\beta(K_0)$ is
defined by $\ell(f)\Phi=\Phi\circ j(f)$ and $\ell(f)^*$ is its adjoint.
Differentiation of $\mu_f$ proceeds according to Proposition 2.1(b).
The function $\ell:{\cal A}\to{\cal L}(C^{r-1}(M),C^\beta(K_0))$ is
$r-2$ times continuously differentiable (as seen by direct computation
because if $\Phi\in C^{r-1}$, its first $r-2$ derivatives are still
$C^1$, which by composition with a $C^\beta$ function gives a $C^\beta$
function). The same holds therefore for
$$ \ell^*:{\cal A}\to{\cal L}(C^\beta(K_0)^*,C^{r-1}(M)^*) $$
We may now differentiate $\ell(f)^*(\mu_f|C^\beta(K_0))$, and we find
that the derivatives up to order $r-2$ are in $C^{r-1}(M)^*$.\qed
\medskip
{\sl 2.3 Remark.}
\medskip
One can probably improve Proposition 2.2 to the statement that
$f\mapsto\rho_f|C^{r-2+\epsilon}(M)$ is
$C^{r-2}$: ${\cal A}\to C^{r-2+\epsilon}(M)^*$ when $\epsilon>0$.
\vfill\eject
\noindent
{\bf 3. Generalized SRB measures: differentiation with respect to $f$.}
\bigskip
\bigskip
For $r\ge3$, we have just seen that $f\mapsto\rho_f=j(f)^*\mu_f$
is $C^1$: ${\cal A}\to C^2(M)^*$. We may thus differentiate this map,
or equivalently compute the tangent map $\delta\rho_f(\Phi)$ to
$$ f\mapsto\rho_f(\Phi)=\mu_f(\Phi\circ j(f)) $$
for $\Phi\in C^2(M)$. The linear functional $\delta
f\mapsto\delta\rho_f(\Phi)$ corresponds to a linear functional
$X\mapsto\delta\rho_f(\Phi)$, where $X=\delta f\circ f^{-1}$ is a
$C^{r-1}$ vector field on $M$. We shall evaluate
$X\mapsto\delta\rho_f(\Phi)$ in two steps.
\medskip
{\sl First step: computing $(\delta\mu_f)(\Phi\circ j(f))$.}
\medskip
By assumption we have the hyperbolic splitting $T_KM=V^s\oplus
V^u$ for $Tf$ over $K$. Let $F=F(f)$ be a section (not necessarily
continuous) of $(V^u)^{\wedge u}$, such that $\|F_x\|=1$ for all $x\in
K$. (We use the norm defined from the Riemann metric; since $(V^u)^{\wedge
u}$ is $1$-dimensional, $F_x$ is unique up to a factor $\pm1$). We have
$$ (T_xf)^{\wedge u}F_x=\lambda(x)F_{fx} $$
$$ |\lambda(x)|=J_f^u(x)\eqno(2) $$
Let now $V^{s\perp}\subset T^*M$ be the subbundle orthogonal to $V^s$.
There is a unique section $F^*=F^*(f)$ of the $1$-dimensional bundle
$(V^{s\perp})^{\wedge u}$ such that $\langle F_x^*,F_x\rangle=1$ for all
$x\in K$. We have
$$ (T_x^*f)^{\wedge u}F_{fx}^*=\lambda(x)F_x^* $$
and
$$ \lambda(x)=\langle F_{fx}^*,(T_xf)^{\wedge u}F_x\rangle $$
\indent
Remember that $f\mapsto x=j(f)x_0$, and $F_x(f)$, $F_x^*(f)$
depend differentiably on $f$. We may thus estimate $\delta J_f^u$ in
terms of $\delta f$ by straightforward first order calculus. [The fact
that $j(f):K_0\to K$ is in general not smooth plays no role here]. It
is convenient to embed $M$ isometrically in ${\bf R}^N$ with the
Euclidean metric (for suitably large $N$). Then $x+T_xM$ may be viewed
as an affine subspace of ${\bf R}^N$, and a local chart of $M$ is
provided by orthogonal projection on $x+T_xM$. Let $|x-y|<\epsilon/10$.
In an $\epsilon$-neighborhood of $x$, the manifolds $M$, $x+T_xM$, and
$y+T_yM$ are $O(\epsilon^2)$-close, and the projections $M\to x+T_xM$, or
$y+T_yM$ preserve distances up to order $\epsilon^2$. This means that
for first order calculations we may consider $M$ as a piece of Euclidean
space near $x$ (or similarly near $fx$), and identify $T_xM$ with $T_yM$.
\medskip
In view of the above considerations we may write, to first order
in $\delta f$,
$$ \delta\lambda(x)=\lambda(x)[\phi(x)-\phi(fx)]
+\langle F_{fx}^*,[\delta(T_xf)^{\wedge u}]F_x\rangle $$
where
$$ \phi(x)=\langle F_x^*,\delta F_x\rangle=-\langle\delta F_x^*,F_x\rangle $$
Note that the arbitrary $\pm1$ factor encountered earlier disappears in
the definition of $\phi(x)$, and that $\phi(\cdot)$ is a continuous function.
\medskip
We have
$$ \delta(T_xf)=T_x(\delta f)=[T_{fx}(\delta f\circ f^{-1})](T_xf) $$
hence
$$ \delta(T_xf)^{\wedge u}=[(1+T_{fx}(\delta f\circ f^{-1}))^{\wedge u}-1]
(T_xf)^{\wedge u} $$
hence
$$ \delta\lambda(x)-\lambda(x)[\phi(x)-\phi(fx)] $$
$$ =\lambda(x)\langle F_{fx}^*,[(1+T_{fx}(\delta f\circ f^{-1}))^{\wedge u}-1]
F_{fx}\rangle=\lambda(x)[\hbox{div}^uX](fx)\eqno(3) $$
where $\hbox{div}^uX$ is the {\it divergence of $X=\delta f\circ f^{-1}$ in
the unstable direction} defined as follows. The orthogonal projection
$M\to x+T_xM$ replaces the vector field $X$ by a function $X':x+T_xM\to
T_xM$. Restriction of $X'$ to $x+V^u(x)$, and projection parallel to
$V^s(x)$ gives a function $X'':x+V^u(x)\to V^u(x)$. Using an orthonormal
basis of $V^u(x)$, we let $\xi_1,\ldots,\xi_u$ be the corresponding
coordinates in $x+V^u(x)$, and $X_1'',\ldots,X_u''$ the corresponding
components of $X''$. It is now readily checked that () holds if we write
$$ \hbox{div}^uX=\sum_{i=1}^u{\partial\over\partial\xi_i}X_i $$
[Note that with our choice of coordinates, the metric tensor may be
considered as constant near $x$; otherwise the expression for
$\hbox{div}^u$ would be more complicated].
\medskip
From (2), and (3) we obtain
$$ \delta[-\log J_f^u\circ j(f)]x_0=-{\delta\lambda(x)\over\lambda(x)} $$
$$ =[-\hbox{div}^uX](fj(f)x_0)+\phi(fj(f)x_0)-\phi(j(f)x_0) $$
or
$$ \delta[-\log J_f^u\circ j(f)]
=[-\hbox{div}^uX]\circ j(f)\circ f_0+\hbox{coboundary} $$
where the coboundary term $\psi\circ f_0-\psi$ does not change the
equilibrium state.
\medskip
Write $\Psi=[-\hbox{div}^uX]\circ j(f)$ so that $\Psi\in
C^\beta(K_0)$. Taking also $\Phi\in C^\beta(K_0)$, we have
$$ (\delta\mu_f)(\Phi)=\sum_{k\in{\bf Z}}
[\mu_f((\Phi\circ f_0^k)\ldotp\Psi)
-\mu_f(\Phi)\ldotp\mu_f(\Psi)] $$
[See [13] Chapter 5, Exercise 5, and use a Markov partition to apply this
result to the present situation]. Finally (with $\Phi\in C^2(M)$)
$$ (\delta\mu_f)(\Phi\circ j(f))=\sum_{k\in{\bf Z}}
[\rho_f((\Phi\circ f^k)\ldotp(-\hbox{div}^uX))
-\rho(\Phi)\ldotp\rho_f(-\hbox{div}^uX)] $$
\vfill\eject
{\sl Second step: computing $\mu_f(\delta(\Phi\circ j(f)))$.}
\medskip
Using Proposition 1.2(c) we have
$$ \delta(\Phi\circ j(f))x_0=\langle T_{j(f)x_0}\Phi,\delta j(f)x_0\rangle
=\langle T_{j(f)x_0}\Phi,(1-{\cal T}_{j(f)})^{-1}
(\delta f\circ f^{-1}\circ j(f))x_0\rangle $$
where
$$ ({\cal T}_{j(f)}(Y\circ j(f))x_0
=(T_{j(f_0^{-1}x_0)}f)(Y\circ j(f)\circ f_0^{-1})x_0 $$
Write again $x=j(f)x_0$, $X=\delta f\circ f^{-1}$, and let
$X(x)=X^s(x)+X^u(x)$ with $X^s(x)\in V^s(x)$, $X^u(x)\in V^u(x)$. We
have then
$$ ({\cal T}_{j(f)}^k(Y\circ j(f))x_0
=(T_{f^{-k}x}f^k)(Y\circ f^{-k})x $$
and
$$ \delta(\Phi\circ j(f))x_0 $$ $$
=\langle T_x\Phi,\sum_{n=0}^\infty{\cal T}_{j(f)}^n(X^s\circ j(f))x_0\rangle-
\langle T_x\Phi,\sum_{n=1}^\infty{\cal T}_{j(f)}^{-n}(X^u\circ j(f))x_0\rangle
$$ $$
=\langle T_x\Phi,\sum_{n=0}^\infty(T_{f^{-n}x}f^n)X^s(f^{-n}x)\rangle
-\langle T_x\Phi,\sum_{n=1}^\infty(T_{f^nx}f^{-n})X^u(f^nx)\rangle
$$ $$
=\sum_{n=0}^\infty\langle T_{f^{-n}x}(\Phi\circ f^n),X^s(f^{-n}x)\rangle
-\sum_{n=1}^\infty\langle T_{f^nx}(\Phi\circ f^{-n}),X^u(f^nx)\rangle $$
\indent
Using the $f_0$-invariance of $\mu_f$, and writing
$\hbox{grad}\,\Phi$ for the element of $T_x^*M$ defined by $T_x\Phi$ we
have thus
$$ \mu_f(\delta(\Phi\circ j(f))) $$
$$ =\int\mu_f(dx_0)
[\sum_{n=0}^\infty\langle T_{j(f)x_0}(\Phi\circ f^n),X^s(j(f)x_0)\rangle
-\sum_{n=1}^\infty\langle T_{j(f)x_0}(\Phi\circ f^{-n}),X^u(j(f)x_0)\rangle] $$
$$ =\rho_f[\sum_{n=0}^\infty\langle\hbox{grad}(\Phi\circ f^n),X^s\rangle
-\sum_{n=1}^\infty\langle\hbox{grad}(\Phi\circ f^{-n}),X^u\rangle] $$
\medskip
{\sl 3.1 Theorem.}
\medskip{\it
Let $K$ be a compact invariant set for the $C^3$ diffeomorphism
$f$ of $M$. We assume that $K$ is hyperbolic with local product
structure and that $f|K$ is mixing. We denote by $\rho_f$ the
generalized SRB state on $K$.
\medskip
(a) The derivative of $f\mapsto\rho_f$ is given by
$$ \delta f\mapsto\delta\rho_f=\delta^{(1)}\rho_f+\delta^{(2)}\rho_f $$
and, for $\Phi\in C^2(M)$,
$$ \delta^{(1)}\rho_f(\Phi)=\sum_{k=-\infty}^\infty
[\rho_f((\Phi\circ f^k)(-{\rm div}^uX^u)-\rho_f(\Phi)\rho_f(-{\rm div}^uX^u)]$$
$$ \delta^{(2)}\rho_f(\Phi)=
\sum_{k=0}^\infty\rho_f\langle{\rm grad}(\Phi\circ f^n),X^s\rangle
-\sum_{k=1}^\infty\rho_f\langle{\rm grad}(\Phi\circ f^{-n}),X^u\rangle $$
where $X^s$, $X^u$ are the components of the vector field $X=\delta
f\circ f^{-1}$ along the stable and unstable subbundles of the
hyperbolic decomposition $T_KM=V^s\oplus V^u$.
\medskip
(b) If $K$ is an attractor, we have $\rho_f({\rm div}^uY)=0$ for
any smooth vector field $Y$, and therefore
$$ \delta\rho_f(\Phi)
=\sum_{n=0}^\infty\rho_f\langle{\rm grad}(\Phi\circ f^n),X\rangle $$
$$=\sum_{n=0}^\infty\rho_f[\langle({\rm grad}\Phi)\circ f^n,(Tf^n)X^s\rangle
-(\Phi\circ f^n){\rm div}^uX^u] $$}
\indent
The proof of (a) has been given above. For (b) we use a Markov
partition and a disintegration of $\rho_f$ into measures carried by
pieces of unstable manifolds. By a change of variable $x\mapsto y=f^Nx$
for $N$ large, and use of Gauss's formula we see that
$\rho_f(\hbox{div}^uY)$ reduces to boundary terms, and since these cancel
pairwise $\rho_f(\hbox{div}^uY)=0$. Therefore $\rho_f(\hbox{div}^uX^u)=0$
and
$$ \rho_f[(\Phi\circ f^k)(-\hbox{div}^uX^u)]
=\rho_f\langle\hbox{grad}(\Phi\circ f^k),X^u\rangle $$
so that
$$ \delta\rho_f(\Phi)
=\sum_{n=0}^\infty\rho_f\langle\hbox{grad}(\Phi\circ f^n),X^s+X^u\rangle $$
as announced.\qed
\medskip
{\sl 3.2 Remarks.}
\medskip
{\it (a)} In the attractor case the formula for
$\delta\rho_f(\Phi)$ contains a term
$$\sum_{n=0}^\infty\rho\langle{\rm grad}\Phi,((Tf^n)X^s)\circ f^{-n}\rangle$$
which converges exponentially because $Tf$ is a contraction on $V^s$,
and a term
$$ \sum_{n=0}^\infty\rho[\Phi\cdotp(({\rm div}^uX^u)\circ f^{-n})] $$
which converges exponentially because of the exponential decay of
correlations for the Gibbs state $\rho$.
\medskip
{\it (b)} Let $m$ be a probability measure absolutely continuous with
respect to Riemann volume on $M$, and with support in the basin of the
attractor $K$. Then $f^{*n}m$ has the weak limit $\rho_f$ when
$n\to\infty$. We may write
$$ \delta[(f^{*n}m)(\Phi)]=\delta m(\Phi\circ f^n)
=\int m(dx)\,\delta\Phi(f^nx) $$
$$=\int m(dx)\langle(\hbox{grad}\Phi)(f^nx),\delta f^nx\rangle $$
$$ =\int m(dx)\langle(\hbox{grad}\Phi)(f^nx),
\sum_{k=0}^{n-1}(Tf^k)\delta f(f^{n-k-1}x)\rangle $$
$$ =\sum_{k=0}^{n-1}\int((f^{n-k})^*m)(dy)\langle(\hbox{grad}\Phi)(f^ky),
(Tf^k)\delta f(f^{-1}y)\rangle $$
$$ =\sum_{k=0}^{n-1}\int((f^{n-k})^*m)(dy)
\langle(\hbox{grad}(\Phi\circ f^k))(y), X(y)\rangle $$
When $n\to\infty$ we obtain formally
$$ \delta\rho_f(\Phi)
=\sum_{k=0}^\infty\rho_f\langle\hbox{grad}(\Phi\circ f^k),X\rangle $$
as asserted in the theorem.
\vfill\eject
\noindent
{\bf 4. Bounded time dependent perturbations.}
\bigskip
\bigskip
Let ${\cal B}_\infty\subset{\cal B}^{\bf Z}$ be the Banach space
of sequences $(X_k)_{k\in{\bf Z}}$ such that
$$ \|(X_k)\|_\infty=\sup_k\|X_k\|<\infty $$
Then, with the notation of Section 1, $B^{\bf Z}\subset{\cal B}_\infty$
($B^{\bf Z}$ contains the open $\epsilon$-ball of ${\cal B}_\infty$).
Note that $0\in B^{\bf Z}$ corresponds to $(K\hookrightarrow M)^{\bf Z}$ and
is a fixed point of the map
$$ (j_k)_{k\in {\bf Z}}\to(f\circ j_{k-1}\circ f^{-1})_{k\in{\bf
Z}} $$
This map is differentiable, and its derivative at 0 is a hyperbolic
linear operator in ${\cal B}_\infty$. Therefore if ${\bf
f}=(f_k)\in{\cal A}^{\bf Z}$, the map
$$ (j_k)_{k\in {\bf Z}}\to(f_k\circ j_{k-1}\circ f^{-1})_{k\in{\bf
Z}} $$
has a unique fixed point ${\bf j}\in B^{\bf Z}$, yielding a diagram
\def\fl#1{\uparrow\vbox to 5mm{}\rlap{$\scriptstyle #1$}}
$$\matrix{
\cdots&\rightarrow&K_{k-1}&\buildrel f_k\over{\rightarrow}&K_k
&\buildrel f_{k+1}\over{\rightarrow}&K_{k+1}&\rightarrow
&\cdots\cr&&\fl{j_{k-1}}&&\fl{j_k}&&\fl{j_{k+1}}&&\cr
\cdots&\rightarrow&K&\buildrel f\over{\rightarrow}&K
&\buildrel f\over{\rightarrow}&K&\rightarrow&\cdots\cr
}$$
where the vertical arrows are the components $j_k$ of ${\bf j}$ and
$K_k=j_kK$. The diagram is commutative because $j_k=f_k\circ
j_{k-1}\circ f^{-1}$. Using the expansiveness of $f$ on $K$, one checks
that the $j_k$ are homeomorphisms. The diagram expresses
structural stability at the level of bounded time dependent
perturbations of a hyperbolic dynamical system.
\medskip
Because the $j_k$ are close to the identity, and the $f_k$ close
to $f$, one can define (un)stable bundles $V_k^\pm$ with the obvious
properties, and (un)stable manifolds ${\cal V}_k^\pm(x)$, such that
$j_k^{-1}{\cal V}_k^\pm(j_kx)$ coincides with ${\cal V}^\pm(x)$ in a
sufficiently small neighborhood of x. The proofs of these facts go
along standard lines, and we do not give them here. We shall now
outline how SRB states can be defined in the present situation where
there is no time stationarity. The proofs will only be sketched.
\medskip
{\sl SRB states.}
\medskip
We first recall the definition of SRB measure in the case of a
single diffeomorphism $f$. Suppose that $K$ is a mixing Axiom A
attractor for $f$, and let $m(dx)=\underline m(x)\,dx$ be a
probability measure absolutely continuous with respect to the
Riemann volume element $dx$, and with support in the basin of attraction
of $K$. Then, when $n\to\infty$, $f^{*n}m$ tends to the SRB measure
$\rho$. One way to see that the limit exists (see [12])
is to choose a Markov partition of $(K,f)$ formed of rectangles
$[S_i,U_i]$. Displacing the mass of $m(dx)$ by a bounded distance along
stable manifolds, we obtain measures $m_i$ on the pieces $U_i$ of
unstable manifolds, where $m_i$ is absolutely continuous with respect to
the Riemann volume element of $U_i$. The weak limit of $f^{*n}m$
remains the same if $m$ is replaced by the sum of the $m_i$, and this
leads to a standard transfer operator study and to the identification of
the limit $\rho$. The SRB state $\rho$ may be characterized in four
different ways:{\it
\medskip
(i) as limit of $f^{*n}m$ where $m$ is absolutely continuous
with respect to $dx$,
\medskip
(ii) as $f$-invariant measure absolutely continuous along
unstable directions,
\medskip
(iii) in terms of eigenfunctions of transfer operators ${\cal
L}$ and ${\cal L}^*$,
\medskip
(iv) by a variational principle.}
\medskip
In the situation of bounded time dependent perturbations as
described above, we can still define SRB states as collections
$(\rho_k)$ where {\it $\rho_k$ is a probability measure on $K_k$ and
$f_k^*\rho_{k-1}=\rho_k$.} We may take as definition the property {\it
\medskip
(i*) for each $k$, $\rho_k=\lim_{n\to\infty}f_k^*\cdots f_{k-n}^*m$.}
\medskip
To prove existence and uniqueness of the SRB states,and study
their properties, we may use the maps $j_k$ and a Markov partition into
rectangles $[S_i,U_i]$ for $(K,f)$. Note in particular that $K_k$ is a
union of sets $j_k[s,U_i]$. Choose now $s_i\in S_i$ and let
$\pi_i:[S_i,U_i]\to[s_i,U_i]$ be the projection. Here is a second
characterization of SRB states:{\it
\medskip
(ii*) for each $k$, the conditional measures $\rho_{k,s,i}$ of
$\rho_k$ with respect to the partition $(j_k[s,U_i])$ are absolutely
continuous with respect to the Riemann volume element on unstable
manifolds. Furthermore the densities $\phi_{i,k}$ of the measures
$(j_k\pi_i j_k^{-1})^*(\rho_k|j_k[S_i,U_i])$ with respect to the unstable volume element
are continuous uniformly in $k$.}
\medskip
The second condition in (ii*) could be replaced by various other
uniformity properties.
\medskip
We write
$$ {\cal L}_k\phi_{k-1}=\phi_k $$
to express that the densities $\phi_{i,k}$ are obtained from the
densities $\phi_{i,k-1}$ by application of a transfer operator ${\cal L}_k$
with coefficients constructed from unstable Jacobians. If $\sigma_k$ is
the collection of measures on the $j_k[s_i,U_i]$ corresponding to the
unstable volume elements, and $\tilde\phi=(\tilde\phi_i)$ is arbitrary,
we have
$$ (\sigma_k,{\cal L}_k\tilde\phi)=(\sigma_{k-1},\tilde\phi) $$
i.e. ${\cal L}_k^*\sigma_k=\sigma_{k-1}$. Here is a third
characterization of SRB states:{\it
\medskip
(iii*)$\qquad\qquad (j_k\pi_ij_k^{-1})^*(\rho_k|j_k[S_i,U_i])
=\phi_k\sigma_k$
\par\noindent
where $\phi_k$ is (up to normalization)
$\lim_{n\to\infty}{\cal L}_k\cdots{\cal L}_{k-n}1$.}
\medskip
The ${\cal L}_k$, acting on a space of H\"older continuous
functions, are close to ${\cal L}$, and there is thus a cone $C$
containing the "principal" eigenvector of ${\cal L}$, and mapped inside
itself by all ${\cal L}_k$. From this one obtains that
${\cal L}_k\cdots{\cal L}_{k-n}1$ converges to a limit $\phi_k$.
\medskip
Adapting for instance the study in [12] to the time dependent
situation, it is now easy to prove existence and uniqueness of SRB
states, and equivalence of {\it (i*), (ii*), (iii*)}. Note that we have
here a situation close to the study of Gibbs states and equilibrium
states by Bogensch\"utz and Gundlach [2], Khanin and Kifer [9], Baladi [1],
where however $(f_k)_{k\in{\bf Z}}$ is distributed according to some
$\tau$-ergodic measure ${\bf P}$. In that case, one obtains only ${\bf
P}$-a.e. statements, but one gains equivalence of {\it (i*), (ii*),
(iii*)} with a variational principle {\it (iv*)}.
\medskip
{\sl Causality.}
\medskip
Note that the "attractors" $K_k$ and the "SRB measures" $\rho_k$
depend only on $f_{k-n}$, $n\ge0$. However, the $j_k$, the
$(j_k\pi_ij_k^{-1})^*(\rho_k|j_k[S_i,U_i])$ and the densities $\phi_k$ depend
on all $f_j$ (because their definitions involve projection along stable
manifolds).
\medskip
{\sl Differentiation of the map ${\bf f}\to \rho_0$.}
\medskip
We shall not embark in a general study of the smoothness of the
map ${\bf f}\to \rho_0$, although such a study should be possible. What
is easy is to vary a finite number of the $f_k$, say those with $|k|\le
N$, because $\rho_{-N}$ then remains fixed, and we have
$$ \rho_0=f_0^*\ldots f_{-N}^*\rho_{-N-1} $$
In particular,
$$ \delta\rho_0(\Phi)=\delta(f_0^*\ldots f_{-N}^*\rho_{-N-1})(\Phi)
=\delta\rho_{-N-1}(\Phi\circ f_0\circ\ldots\circ f_{-N}) $$
$$=\sum_{n=0}^N\rho_{-N-1}(T(\Phi\circ f_0\ldots\circ f_{-n+1})
\delta f_n\circ f_{-n-1}\circ\ldots\circ f_{-N}) $$
$$ =\sum_{n=0}^N\int\rho_{-N-1}(dx)\,
\langle{\rm grad}_{f_{-n}\ldots f_{-N}x}(\Phi\circ f_0\ldots\circ f_{-n+1}),
(\delta f_{-n}\circ f_{-n}^{-1})(f_{-n}\ldots f_{-N}x)\rangle $$
$$ =\sum_{n=0}^N(f_{-n}^*\ldots f_{-N}^*\rho_{-N-1})
\langle{\rm grad}(\Phi\circ f_{-1}\ldots\circ f_{-n}),X_{-n}\rangle $$
where $X_k$ is the vector field $\delta f_k\circ f_k^{-1}$.
\medskip
Finally, we have thus
$$ \delta\rho_0(\Phi)=\sum_{n=0}^\infty\rho_{-n}
\langle{\rm grad}(\Phi\circ f_0\ldots\circ f_{-n+1}),X_{-n}\rangle $$
$$ =\sum_{n=0}^\infty\rho_0\langle{\rm grad}\Phi,
(T(f_0\circ\cdots\circ f_{-n+1})X_{-n}^s)
\circ(f_0\circ\cdots\circ f_{-n+1})^{-1}\rangle $$
$$ -\sum_{n=0}^\infty\rho_0[\Phi\cdotp(({\rm div}^uX_{-n}^u)
\circ(f_0\circ\cdots\circ f_{-n+1})^{-1})] $$
Note that this is formally identical with the result of theorem 3.1(b)
when we replace $\rho_k$ by $\rho$ and $f_k$ by $f$.
\vfill\eject
\noindent
{\bf 5. Formal derivative of ${\bf\rho}_f$ in the general case.}
\bigskip
\bigskip
We assume that the $f$-invariant state $\rho$ satisfies the SRB
condition, but here we do not suppose uniform hyperbolicity, ({\it
i.e.}, ${\rm supp}\rho$ need not be a hyperbolic invariant set). Thus
we do not know how $\rho$ will vary with $f$, but we have a good formal
candidate for its derivative, {\it viz.},
$$ \delta\rho(\Phi)
=\sum_{n=0}^\infty\rho\langle{\rm grad}(\Phi\circ f^n),X\rangle $$
where $X=\delta f\circ f^{-1}$. If there are no vanishing Lyapunov
exponents, a measurable splitting $T_xM=V^s(x)\oplus V^u(x)$ is defined
$\rho(dx)$-a.e., and we may write $X(x)=X^s(x)+X^u(x)$ with
$X^s(x)\in V^s(x)$, $X^u(x)\in V^u(x)$. Then
$$ \rho\langle{\rm grad}(\Phi,f^n),X\rangle
=\rho\langle{\rm grad}(\Phi,f^n),X^s+X^u\rangle $$
$$ =\rho\langle({\rm grad}\Phi)\circ f^n,(Tf^n)X^s\rangle
-\rho((\Phi\circ f^n)\cdotp{\rm div}^uX^u) $$
with $\rho({\rm div}^uX^u)=0$ just as in the uniformly hyperbolic case.
Formally, we have thus
$$ \delta\rho(\Phi)
=\sum_{n=0}^\infty\rho\langle({\rm grad}\Phi)\circ f^n,(Tf^n)X^s\rangle
-\sum_{n=0}^\infty\rho((\Phi\circ f^n)\cdotp{\rm div}^uX^u) $$
The convergence of the right-hand side depends on how $(Tf^n)X^s$ and
$\rho((\Phi\circ f^n)\cdotp{\rm div}^uX^u)$ tend to 0 when $n\to\infty$.
\medskip
In the time dependent case, the formula becomes
$$ \delta\rho_0(\Phi)
=\sum_{n=0}^\infty\rho_{-n}\langle{\rm grad}(\Phi,f_0\circ\ldots\circ f_{-n+1}),
X_{-n}\rangle $$
where $X_k=\delta f_k\circ f_k^{-1}$. In particular, if all $f_k$ are
equal to $f$ and the $\rho_k$ to $\rho$, we obtain
$$ \delta\rho_0(\Phi)
=\sum_{n=0}^\infty\rho\langle{\rm grad}(\Phi,f^n),X_{-n}\rangle $$
$$ =\sum_{n=0}^\infty\rho\langle({\rm grad}\Phi)\circ f^n,
((Tf^n)X_{-n}^s)\rangle
-\sum_{n=0}^\infty\rho((\Phi\circ f^n)\cdotp{\rm div}^uX_{-n}^u) $$
\indent
There are similar formulae for flows. Suppose for instance that
the state $\rho$ satisfies the SRB condition for the flow $(f^t)$
corresponding to the vector field ${\cal X}$. Let $X_t$ be a time
dependent perturbation of ${\cal X}$, then the derivative of $\rho$ at
time 0 is given formally by
$$ \delta\rho_0(\Phi)=\int_0^\infty dt\,\rho
\langle{\rm grad}(\Phi\circ f^t),X_{-t}\rangle $$
$$ =\int_0^\infty dt\,\rho
\langle({\rm grad}\Phi)\circ f^t,(Tf^t)X_{-t}^s\rangle $$
$$ -\int_0^\infty dt\,\rho((\Phi\circ f^t)({\rm div}^uX_{-t}^u)) $$
\vfill\eject
\noindent
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