Content-Type: multipart/mixed; boundary="-------------1001300835918" This is a multi-part message in MIME format. ---------------1001300835918 Content-Type: text/plain; name="10-27.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="10-27.keywords" dynamical systems, susceptibility, non-equilibrium, SRB measure, stable-unstable tangency. ---------------1001300835918 Content-Type: application/x-tex; name="susceptibility.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="susceptibility.tex" \magnification=1200 \centerline {SINGULARITIES OF THE SUSCEPTIBILITY OF AN SRB MEASURE} \centerline {IN THE PRESENCE OF STABLE-UNSTABLE TANGENCIES. \footnote{*}{An early version of this work was presented at the France-Brazil mathematical conference at IMPA in Sep. 2009, at the LAWNP conference in Buzios in Oct. 2009, and at the meeting {\it Progress in Dynamics} at the IHP in Nov. 2009.}} \bigskip\bigskip \centerline{by David Ruelle\footnote{$\dagger$}{Math. Dept., Rutgers University, and IHES, 91440 Bures sur Yvette, France. email: ruelle@ihes.fr}.} \bigskip\bigskip\bigskip\bigskip\noindent {\leftskip=2cm\rightskip=2cm\sl Abstract. Let $\rho$ be an SRB (or ``physical''), measure for the discrete time evolution given by a map $f$, and let $\rho(A)$ denote the expectation value of a smooth function $A$. If $f$ depends on a parameter, the derivative $\delta\rho(A)$ of $\rho(A)$ with respect to the parameter is formally given by the value of the so-called susceptibility function $\Psi(z)$ at $z=1$. When $f$ is a uniformly hyperbolic diffeomorphism, it has been proved that the power series $\Psi(z)$ has a radius of convergence $r(\Psi)>1$, and that $\delta\rho(A)=\Psi(1)$, but it is known that $r(\Psi)<1$ in some other cases. One reason why $f$ may fail to be uniformly hyperbolic is if there are tangencies between the stable and unstable manifolds for $(f,\rho)$. The present paper gives a crude, nonrigorous, analysis of this situation in terms of the Hausdorff dimension $d$ of $\rho$ in the stable direction. We find that the tangencies produce singularities of $\Psi(z)$ for $|z|<1$ if $d<1/2$, but only for $|z|>1$ if $d>1/2$. In particular, if $d>1/2$ we may hope that $\Psi(1)$ makes sense, and the derivative $\delta\rho(A)=\Psi(1)$ has thus a chance to be defined.\par} \vfill\eject \noindent {\bf 0. Introduction.} \medskip Let $f$ be a diffeomorphism of the compact manifold $M$, and $\rho$ an SRB measure\footnote{$^1$}{For a discussion of SRB measures (Sinai-Ruelle-Bowen) see for instance [10], [27], [2], and references given there. For recent work analyzing SRB measures for a class of noninvertible maps, see [1].} for $f$. The derivative $\delta_X\rho(A)$ of the map $f\mapsto\rho$ in the direction of the smooth vector field\footnote{$^2$}{If we replace $f:x\mapsto fx$ by $x\mapsto fx+X(fx)$, then $\rho$ is replaced by $\rho+\delta_X\rho$ to first order in $X$. The derivative of $f\mapsto\rho$ in the direction of $X$, evaluated at $A$, is $\delta_X\rho(A)$.} ${\bf X}$, evaluated at the smooth real function $A$, can be formally computed to be the value at $z=1$ of $$ \Psi(z)=\sum_{n=0}^\infty z^n\int\rho(dx)\,{\bf X}(x)\cdot\partial_x(A\circ f^n) \eqno{(1)} $$ We shall call $\Psi$ the {\it susceptibility}\footnote{$^3$}{The physical susceptibility is defined for a continuous time dynamical system, and is a function of the frequency $\omega$. For the discrete time dynamics considered here, the susceptibility would be $\omega\mapsto\Psi(e^{i\omega})$, but for simplicity we call $\Psi$ the susceptibility.}. \medskip In the uniformly hyperbolic case (i.e., if the support of $\rho$ is a mixing Axiom A attractor for $f$), $\Psi$ has a radius of convergence $r(\Psi)>1$. One can furthermore prove that $f\mapsto\rho$ is differentiable and that its derivative is given by $\Psi(1)$\footnote{$^4$}{The differentiability of $f\mapsto\rho$ has been established in [9], the inequality $r(\Psi)>1$ and the identity $\delta_X\rho(A)=\Psi(1)$ are proved in [16]. There are corresponding results for hyperbolic flows [18], [3], and generalizations to partially hyperbolic systems [5].}. In nonuniformly hyperbolic situations these assertions may fail: $r(\Psi)$ may be $<1$, and $f\mapsto\rho$ is presumed to be nondifferentiable\footnote{$^5$}{$r(\Psi)<1$ has been proved for certain (noninvertible) unimodal maps of the interval [17], [8], see also [19] and work in progress by Baladi and Smania. The analysis in [19] strongly suggests that for a certain class of unimodal maps, the function $f\mapsto\rho$ is nondifferentiable, even in the (weak) Whitney sense. There is also numerical evidence [4] that $r(\Psi)<1$ for the classical H\'enon attractor. For recent work on H\'enon-like diffeomorphisms, see [14].}. \medskip The above results suggest two problems: I. proving that $r(\Psi)<1,\ge1,$ or $>1$ in cases of some generality, and II. relating the derivative of $f\mapsto\rho$ to $\Psi(1)$ when this quantity is defined. The present note is about the first problem, and presents a nonrigorous study of the singularities of $\Psi$ which may occur as a result of {\it tangencies}, i.e., tangencies of stable and unstable manifolds for the system $(f,\rho)$, assumed to have no zero Lyapunov exponent. (The existence of tangencies excludes uniform hyperbolicity). Our study is not rigorous, but suggests that $r(\Psi)<1$ if the partial Hausdorff dimension $d$ of $\rho$ in the stable direction is $<{1\over2}$, while $r(\Psi)\ge1$ if $d\ge1/2$. This opens the possibility that, for some {\it fat} tangencies ($d$ sufficiently large), $\Psi(1)$ is well defined. In that case, a derivative of $f\mapsto\rho$ may exist, with applications to the physical theory of {\it linear response}\footnote{$^6$}{A basic physical article on linear response is [21]. A review of linear response for dynamical systems is given in [20].}. \medskip\noindent {\bf Acknowledgments.} \medskip I am indebted to Artur \`Avila, Viviane Baladi, Bruno Cessac, Jean-Pierre Eckmann, and Jean-Christophe Yoccoz for valuable remarks. \medskip\noindent {\bf 1. Example: volume preserving diffeomorphisms.} \medskip Let $\ell$ be equivalent to Lebesgue measure on $M$, and let the $f$-invariant probability measure $\rho$ be the restriction of $\ell$ to a certain open set $S\subset M$ [similarly, we may also consider the situation where $f$ acts on ${\bf R}^m$ , and $S$ is a bounded open set in ${\bf R}^m$]. If either supp$X\subset S$, or supp$A\subset S$, we may write $$ \Psi(z) =\sum_{n=0}^\infty z^n\int_S\ell(dx)\,{\bf X}(x)\cdot\partial_x(A\circ f^n) =-\sum_{n=0}^\infty z^n\int_S\ell(dx)\,[{\rm div}_\ell{\bf X}(x)]A(f^nx) $$ and therefore $r(\Psi)\ge1$. \medskip If we furthermore suppose that either supp$X\subset S$, or supp$A\subset S$ and $\ell(A)=0$, we have $$ (f,\rho)\,{\rm exponentially}\,{\rm mixing}\quad \Rightarrow\quad r(\Psi)>1 $$ Volume preserving Anosov diffeomorphisms satisfy this condition, and the same is true of the time 1 map of an exponentially mixing volume preserving Anosov flow (which is not uniformly hyperbolic). Can exponential mixing happen for non-Anosov area preserving diffeomorphisms in 2 dimensions? We shall now see that mixing already implies that $\Psi(1)$ is well defined, and $\delta_X\rho(A)=\Psi(1)$ when the derivative $\delta_X$ is taken along diffeomorphisms preserving a (parameter dependent) volume. \medskip For simplicity we discuss the case $S=M$. Let $\rho$ be a probability measure equivalent to Lebesgue measure on the compact manifold $M$. Denote by $\tilde\rho$ the density of $\rho$ with respect to Lebesgue measure on some charts. Thus $(f^*\rho)^\sim=\tilde\rho\,\circ f^{-1}/J\circ f^{-1}$ where $J(x)=|{\rm det}(D_xf)|$. Suppose now that $f,\tilde\rho$ depend smoothly on a parameter, and denote the derivative with respect to the parameter by a {\it prime}. In particular $f'=X\circ f, J'(x)=J(x)[{\rm div}X(fx)]$. \medskip Writing $\rho_1=f^*\rho$ we have $\tilde\rho_1=(\tilde\rho/J)\circ f^{-1}$, or $\tilde\rho(x)=J(x)(\tilde\rho_1(fx))$, hence $$ \tilde\rho'(x)=J(x)[\tilde\rho_1'(fx)+\partial_{fx}\tilde\rho_1\cdot X(fx)] +J(x)[{\rm div}X(fx)]\tilde\rho_1(fx) $$ or $$ (\tilde\rho'/J)\circ f^{-1} =\tilde\rho_1'+\partial\tilde\rho_1\cdot X+[{\rm div}X]\tilde\rho_1 =\tilde\rho_1'+{\rm div}(\tilde\rho'_1X) =\tilde\rho_1'+[{\rm div}_{\tilde\rho_1}X]\tilde\rho_1 $$ hence $$ \int dx\,\tilde\rho'(x)A(fx)=\int dx\,\tilde\rho_1'(x)A(x) +\int dx\,\tilde\rho_1(x)[{\rm div}_{\tilde\rho_1}X(x)]A(x) $$ Imposing the invariance condition $\rho=f^*\rho$, we have thus $$ \int dx\,\tilde\rho'(x)A(f^{N+1}x)=\int dx\,\tilde\rho'(x)A(x) +\sum_{n=0}^N\int dx\,\tilde\rho(x)[{\rm div}_{\tilde\rho}X(x)]A(f^nx)\eqno{(2)} $$ Note that $\int dx\,\tilde\rho(x)=1$ implies $\int\rho(dx)\,(\tilde\rho'/\tilde\rho)(x)=\int dx\,\tilde\rho'(x)=0$. Therefore, imposing mixing gives that $$ \int dx\,\tilde\rho'(x)A(f^{N+1}x) =\int\rho(dx)\,(\tilde\rho'/\tilde\rho)(x)A(f^{N+1}x) $$ tends to 0 when $N\to\infty$. Equation $(2)$ now implies that $$ \Psi(z)=-\sum_{n=0}^\infty z^n \int dx\,\tilde\rho(x)[{\rm div}_{\tilde\rho}X(x)]A(f^nx) $$ is well defined for $z=1$, and $\int dx\,\tilde\rho'(x)A(x)=\Psi(1)$. \medskip {Conclusion: }{\sl Suppose that $\rho$ is $f$-ergodic, with density $\tilde\rho$, and that $(f,\rho)$ is mixing on a function space ${\cal S}$ containing $\tilde\rho'/\tilde\rho$ and $A$, then $\Psi(1)$ is well defined, and $\delta_X\rho(A)=\int dx\,\tilde\rho'(x)A(x)=\Psi(1)$.} \medskip\noindent {\bf 2. Computer simulations.} \medskip It is accepted that, using a computer, one can approximate numerically an SRB measure by a time average: $$ {1\over N_1-N_0}\sum_{n=N_0+1}^{N_1}\delta_{f^nx} $$ for large $N_1-N_0$ (and $N_0$ moderately large); the idea is that the computed orbit $f^nx$ is noisy because of roundoff errors, and that this noisy orbit has an SRB time average\footnote{$^7$}{See [15] and, for example, Eckmann and Ruelle [6].}. The Lyapunov exponents, and the coefficient $L_+$ introduced below, can also in principle be determined numerically. It is therefore possible to estimate $r(\Psi)$ in particular cases, and to test the relations proposed above between the stable dimension $d$ of $\rho$ and the convergence radius $r(\Psi)$ in the presence of tangencies. For example, let dim $M=2$, and let the Lyapunov exponents $\lambda_-,\lambda_+$ of $(f,\rho)$ satisfy $\lambda_-<0<\lambda_+$, so that\footnote{$^8$}{See L.-S. Young [24].} $d=\lambda_+/|\lambda_-|$. Does the presence of tangencies together with $\lambda_+/|\lambda_-|<1/2$ imply $r(\Psi)<1$? (This appears to be the case for the classical H\'enon attractor). Does $\lambda_+/|\lambda_-|\ge1/2$ imply $r(\Psi)\ge1$? Are there examples with tangencies and $r(\Psi)>1$? \medskip\noindent {\bf 3. Singularities of $\Psi$ in the presence of tangencies.} \medskip It is readily seen that the power series (1) defining the susceptibility has a radius of convergence $r(\Psi)>0$. Tangencies between stable and unstable manifolds for $(f,\rho)$ are expected to produce singularities of $\Psi$, thus limiting $r(\Psi)$. A difficulty of the problem is that the set of points of tangency has measure zero. Note in this respect that the angle between stable and unstable manifolds is defined $\rho$-a.e., and that the a.e. range of this angle determines if tangencies are allowed or not. A similar comment can be made for higher order contacts of the stable-unstable manifolds. It is reasonable to exclude those higher order contacts which (given the dimension of $M$) are nongeneric if the stable and unstable manifolds are regarded as independent. At a generic tangency point $O$, the unstable manifold is folded in a way which is basically 2-dimensional (corresponding to variables $x,y$ introduced below). Along the orbit $(f^nO)$ we have folds which are sharper and sharper as $n\to\infty$. This exponential sharpening of the folds, combined with the derivative $\partial_x$ in $(1)$, produces the singularities of $\Psi(z)$ with $|z|<1$ which we want to study. \medskip One can prove that $r(\Psi)<1$ for certain unimodal maps of the interval\footnote{$^{9}$}{See footnote 5.}; these maps are non-invertible and give a degenerate example of tangencies that is relatively accessible to mathematical study. In what follows we discuss a crude imitation of the 1-dimensional situation for higher-dimensional diffeomorphisms. In the case of unimodal maps of the interval with an ergodic measure $\rho$ absolutely continuous with respect to Lebesgue, the density of $\rho$ has {\it spikes} $\sim|x-f^nc|^{-1/2}$ on one side of the points $f^nc$ of the postcritical orbit. These spikes are at the origin of the singularities of $\Psi(z)$ inside of the unit circle. Instead of an individual postcritical point, we find for higher dimensional diffeomorphisms a family of tangencies of stable and unstable manifolds: think of a pile of (local) unstable manifolds (with tangencies) carrying part of the measure $\rho$. Morally, this means that the spikes are ``spread out'' or ``smoothed'' (corresponding to integration over a measure transverse to the unstable manifolds). This smoothing may give weaker singularities of $\Psi$ (i.e., larger $r(\Psi)$). \medskip Let us choose local coordinates $(x,X,y,Y)\in{\bf R}\times{\bf R}^{s-1}\times{\bf R}\times{\bf R}^{u-1}$ such that the $s$-dimensional stable manifolds are given by $(y,Y)=$ const., and the local unstable manifold $U$ through $O$ is given by $x=ay^2,X=0,Y=0$ (for definiteness we take $a>0$). The conditional measure of $\rho$ on $U$ is thus $\sim\Delta(dx\,dX\,dy\,dY)=\delta(x-ay^2)\delta(X)\delta(Y)dx\,dX\,dy\,dY$. One can argue that the variable $Y$ does not play an important role in the present discussion, and we shall omit it, which amounts to taking $u=1$. Using similar local coordinates near $fO$, we assume that the map $f$ has the form $$ (x,X,y)\mapsto(e^{L_+}x,e^\Lambda X,e^{L_-}y) $$ where $L_-<0,L_+>0$, and $e^\Lambda$ is a contraction (stronger than that given by $e^{L_-}$). \medskip The assumption that the unstable manifolds are parallel affine manifolds is crude, and so is the assumption that $L_-,L_+$, and $\Lambda$ are constant coefficients. [One might think of $L_-,L_+$, and $\Lambda$ as Lyapunov exponents. But $L_+$, the only one of these coefficients to appear in the final formulas, is really the mean rate of expansion along a forward orbit $(f^nO)$ of tangencies]. These crude assumptions may be in part justified by the fact that we are looking for the leading singular behavior associated with a subset of unstable manifolds. We shall use informally the notation $\approx$ (approximately equal to) and $\sim$ (approximately proportional to) in trying to find the leading singular behavior. \medskip The contribution of the conditional measure $\Delta$ of $\rho$ on the piece $U$ of unstable manifold is $$ \Psi^\Delta(z)\sim\sum_{n=0}^\infty z^n \int\Delta(dx\,dX\,dy)\,{\bf X}(x,X,y)\cdot\partial_{(x,X,y)}(A\circ f^n) $$ Singularities for $|z|\le1$ can only come from the component ${\bf X}_1$ of ${\bf X}$ in the $x$-direction, giving $$ \Psi^\Delta(z)\sim\sum_{n=0}^\infty z^n \int dx\,dy\,\delta(x-ay^2){\bf X}_1\partial_x(A_1\circ f^n) $$ $$ \approx\sum_{n=0}^\infty(ze^{L_+})^n{\bf X}_1(0) \int (f^{*n}\delta(x-ay^2)dx\,dy)A'_1(x) $$ where $A'_1(x)$ is the derivative of $A_1(x)$, which is $A$ evaluated in the coordinates $(x,0,0)$ centered at $f^nO$. \medskip One has $$ \delta(x-ay^2)={1\over2\sqrt{ax}} [\delta(y-{\sqrt{x}\over\sqrt{a}})+\delta(y+{\sqrt{x}\over\sqrt{a}})] $$ hence $$ f^{*n}(\delta(x-ay^2)dx\,dy) ={e^{-nL_+/2}\over2\sqrt{ax}}[\delta(y-{\sqrt{x}\over\sqrt{a_n}}) +\delta(y+{\sqrt{x}\over\sqrt{a_n}})]dx\,dy $$ where $a_n=ae^{nL_+}e^{-2nL_-}$. Therefore $$ \Psi^\Delta(z)\approx \sum_{n=0}^\infty(ze^{L_+/2})^n {{\bf X}_1(0)\over\sqrt{a}}\int_0^{\rm cutoff}{dx\over\sqrt{x}}A'_1(x) $$ so that $r(\Psi^\Delta)=e^{-L_+/2}<1$. This result is in agreement with that obtained with the spikes of the invariant density for unimodal maps in 1 dimension (which are limiting cases of diffeomorphisms with tangencies). \medskip Remember however that $\Psi$ is defined with the measure $\rho$ rather than $\Delta$. Let thus $\Gamma$ be part of the measure $\rho$, carried by a pile of unstable manifolds (with tangencies) near $O$, and write $$ \Gamma(dx\,dX\,dy) =\int\gamma(d\xi\,dX)\,\delta(x-a(\xi,X)(y-b(\xi,X))^2-c(\xi,X))dx\,dy$$ where the integration is over the variable $\xi$, and $\gamma(d\xi\,dX)$ is a transverse measure of $\rho$ in the stable direction, and we assume $a(\xi,X)>0$. It will turn out that we obtain the same estimate of $r(\Psi^\Gamma)$ for different $\Gamma$'s, and we expect that the contributions $\Psi^\Gamma$ to $\Psi$ of different $\Gamma$'s will add up convergently for $|z|0$, we have thus $h\in L_p$ (and the bound $p<(1-d)/(1/2-d)$ appears best possible). Let $1/p+1/q=1$, then the Fourier transform $\hat h$ is in $L_q$. We have $$ {1\over q}<1-{1/2-d\over1-d}={1/2\over1-d}\eqno{(3)} $$ (and this bound appears best possible). If $|\hat h(s)|\sim s^{-t}$ for large $s$, we need $tq>1$, i.e., $t>1/q$ if $1/q$ satisfies $(3)$, i.e., $$ |\hat h(s)|\sim s^{-t}\qquad{\rm with}\qquad t\ge{1/2\over1-d} $$ We come now to the estimation of $\int h(x)A_1'(e^{nL_+}x)\,dx$ where, for large $n$, $x\mapsto A_1'(e^{nL_+}x)$ is rapidly oscillating with frequency $\sim nL_+$. Since we are interested in the most singular part of $h$, we may replace it by a function with compact support. Because $A'_1$ is a derivative, there is no zero-frequency contribution to the integral, and we have $$ \int h(x)A_1'(e^{nL_+}x)\,dx\sim\hat h(e^{nL_+}) $$ with a negligible contribution of higher harmonics. Therefore $$ |\int h(x)A'_1(e^{nL_+}x)\,dx|\sim|\hat h(e^{nL_+})| \sim e^{-tnL_+}\le\exp[-n{1/2\over1-d}L_+] $$ and the bound again appears best possible, so that $\Psi_1^\Gamma(z)$ converges for $$ |z|<\exp[-(1-{1/2\over1-d})L_+] =\exp[-{1/2-d\over1-d}L_+] $$ i.e., $r(\Psi_1^\Gamma)=\exp[-(1/2-d)L_+/(1-d)]$, and a reasonable guess would appear to be $$ r(\Psi)=\exp[-{1/2-d\over1-d}L_+] $$ [hence $e^{-L_+/2}0,\alpha+\beta=1/2,d+\alpha<1$, or $\beta=1/2-\alpha,0<\alpha<1-d$. We have thus $$ h=h_1*(\cdot)^{\beta-1}\qquad{\rm where} \qquad h_1=\tilde\gamma*(\cdot)^{\alpha-1} $$ and we have seen that $h_1(x)\approx C_\alpha(x-\xi^*)^{d+\alpha-1}$. We find as in {\bf A.} that we can take $|\hat h_1(s)|\sim s^{-t}$ with $t\ge\alpha/(1-d)$, hence $$ \hat h(s)\sim s^{-\alpha/(1-d)}s^{-\beta} =s^{-1/2-\alpha d/(1-d)} $$ so that $$ |\int h(x)A_1'(e^{nL_+}x)\,dx|\sim|\hat h(e^{nL_+})| \le\exp[-n({\alpha d\over1-d}+{1\over2})L_+] $$ and $\Psi_1^\Gamma(z)$ converges for $$ |z|<\exp[({\alpha d\over1-d}+{1\over2}-1)L_+] =\exp[({\alpha d\over1-d}-{1\over2})L_+] $$ where we may let $\alpha\to1-d$, hence we may estimate $$ r(\Psi_1^\Gamma)\ge\exp[({\alpha d\over1-d}-{1\over2})L_+] =\exp[(d-1/2)L_+] $$ which is $>1$. In fact $r(\Psi_1^\Gamma)=\exp[(d-1/2)L_+]$ is a reasonable guess. \medskip The convergence radius $r(\Psi)$ now depends on the behavior of $(f,\rho)$ away from tangencies, and we may expect that the derivative $\partial_x$ in $(1)$ plays a less important role. Therefore $r(\Psi)$ is expected to depend on the mixing properties of $(f,\rho)$, over which one has some control [25], [26]. One may thus hope that $r(\Psi)\ge1$, or even $r(\Psi)>1$, and that $\Psi(1)$ is well defined. The situation where the set of tangencies is large ($d>1/2$) reminds one of Newhouse's study of persistent tangencies (wild hyperbolic sets, infinitely many sinks, see [11], [12], [13]). While the situation considered by Newhouse has very discontinuous topology, it is not unthinkable that the particular measure $\rho$ behaves differentiably in some sense. \medskip\indent {\bf C.} It is plausible that the results of {\bf A.} and {\bf B.} remain true without the condition that supp$\tilde\gamma$ has zero Lebesgue measure. Furthermore, if the stable dimension $d$ of $\rho$ is $\ge1$, one can write $d$ as a sum of partial dimensions\footnote{$^{10}$}{See [6] Section IV.D, and references given there, in particular [10].}, and use arguments as above. One expects thus that the formula $r(\Psi_1^\Gamma)\ge\exp[(d-1/2)L_+]$ will remain correct in that case and, as argued in {\bf B.}, we may then have $r(\Psi)\ge1$ or even $r(\Psi)>1$. \medskip If we have a continuous time dynamical system (a flow) instead of discrete time dynamics (a diffeomorphism), we expect similar results in the presence of tangencies: a susceptibility function $\hat\kappa(\omega)$ with singularities in the upper half $\omega$-plane if $d<1/2$, no singularity if $d\ge1/2$, and $\hat\kappa(0)$ hopefully well defined if $d>1/2$. The continuous time dynamical situation is that most relevant for physical applications. \medskip\noindent {\bf 5. Physical discussion.} \medskip In this brief physically oriented discussion we shall, for simplicity, use the language of discrete time dynamical systems. \medskip We have made above a nonrigorous analysis of how tangencies between stable and unstable manifolds may influence the radius of convergence $r(\Psi)$ of the susceptibility function. We have found two different regimes depending on whether the stable dimension $d$ of the SRB measure $\rho$ is $<1/2$ or $\ge1/2$. \medskip If $d<1/2$ we expect $r(\Psi)<1$, i.e., the tangencies cause singularities of $\Psi(z)$ with $|z|<1$. Such singularities reflect the exponential growth of small periodic perturbations of the dynamics $(f,\rho)$ (see [20]). Experimentally, this may be visible as resonant behavior when a physical system is excited by a weak periodic signal: it would be of particular interest to study the case of hydrodynamic turbulence. \medskip If $d\ge1/2$ we expect $r(\Psi)\ge1$ and, if $d>1/2$, the value $\Psi(1)$ may be well defined. Since $\Psi(1)$ is formally related to the derivative of $\rho$ with respect to $f$, we may hope that this derivative exists in some sense. This would apply to physical systems not too far from equilibrium (at equilibrium, $\rho$ has a density, and $d\ge1$ unless all Lyapunov exponents vanish) with obvious application to linear response in nonequilibrium statistical mechanics. For large physical systems (dim$M$ large), when there is chaos and a density of Lyapunov exponents can be defined, one also expects $d$ large by the Kaplan-Yorke formula\footnote{$^{11}$}{See [6] Section IV.C, and references given there, in particular [7].}, provided the degrees of freedom of the large system have a sufficiently strong effective interaction. \medskip In view of the mathematical difficulty of analyzing dynamical systems with tangencies, a computer-experimental study would be desirable. The situation of choice would be that of 2-dimensional diffeomorphisms with an SRB measure $\rho$ such that the Lyapunov exponents $\lambda_-,\lambda_+$ satisfy $\lambda_-<0<\lambda_+$. In that case we know [24] that $d=\lambda_+/|\lambda_-|$, and the radius of convergence $r(\Psi)$ is also accessible numerically. For the classical H\'enon attractor we have $d<1/2$, and it appears [4] that $r(\Psi)<1$. In other cases, studied by Ueda and coworkers [22], [23], visual inspection of the computer plot of the attractor seems to indicate a large $d$, and it would be desirable to estimate $r(\Psi)$. \medskip\noindent {\bf 6. Infinitesimally stable ergodic measures.} \medskip Consider the general situation of a diffeomorphism $f$ of the compact manifold $M$, and of an ergodic measure $\rho$ for $f$ on $M$. We want to study formally the stability of $\rho$ under an infinitesimal change of $f$. \medskip We shall use a space ${\cal D}$ of smooth functions on $M$, with dual ${\cal D}^*$, and a space ${\cal V}$ of smooth vector fields on $M$. If $X\in{\cal V}$, we write $\hat X(A)=\int\rho(dx)\,X(x)\cdot\partial_xA$, so that $\hat X\in{\cal D}^*$. Defining $T:{\cal D}^*\to{\cal D}^*$ and $Tf:{\cal V}\to{\cal V}$ by $$ (T\xi)(A)=\xi(A\circ f)\qquad,\qquad((Tf)X)(fx)=(T_xf)X(x) $$ we find that $((Tf)X)^\wedge=T\hat X$. \medskip Consider $\rho+\hat X$ as an infinitesimal perturbation of $\rho$ (it corresponds to replacing $\rho$ by its image under $x\mapsto x+X(x)$). The measure $\rho$ is mapped to itself by $f$, while $\rho+\hat X$ is mapped to $\rho+T\hat X$. We say that $\rho$ is {\it infinitesimally stable} (or attracting) if $(T^n\hat X)(A)\to0$ exponentially\footnote{$^{12}$}{An alternate (weaker) requirement would be that $\Psi(1)=\sum_{n=0}^\infty\rho(dx)\,X(x)\cdot\partial_x(A\circ f^n)$ converges whenever $X\in{\cal V},A\in{\cal D}$.} with $n$ whenever $X\in{\cal V},A\in{\cal D}$. It is plausible that an infinitesimally stable measure must be SRB. \medskip We perturb $f$ to $\tilde f=f+X\circ f$, where $X\in{\cal V}$. If $\xi\in{\cal D}^*$, the $\tilde f$-invariance of $\rho+\xi$, i.e., $$ (\rho+\xi)(A\circ(f+X\circ f))=(\rho+\xi)(A) $$ is then given, to first order in $X$, by $$ \int\rho(dx)\,[A(fx)+X(fx)\cdot\partial_{fx}A]+\xi(A\circ f)=\rho(A)+\xi(A) $$ or $\hat X+T\xi=\xi$, hence $T^n\xi-T^{n+1}\xi=T^n\hat X$, hence $\xi-T^{N+1}\xi=\sum_{n=0}^NT^n\hat X$. Therefore, if $\rho$ is infinitesimally stable, we obtain $\rho+\xi$ which is $\tilde f$-invariant to first order by taking $$ \xi(A)=\sum_{n=0}^\infty(T^n\hat X)(A) $$ and $\xi$ is unique such that $(T^n\xi)(A)\to0$ for all $A\in{\cal D}$ when $n\to\infty$. This shows that the linear response $X\mapsto\xi$ is related to infinitesimal stability. \medskip The above considerations apply to the uniformly hyperbolic situation where $\rho$ is an SRB measure on an Axiom A attractor. 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