Singularities of the susceptibility of an SRB measure in the presence of stable-unstable tangencies

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

12 pages

References in corpus (1)

• A review of linear response theory for general differentiable dynamical systems

Cited by in corpus (4)

• Linear response, or else
• A mechanical model for Fourier's law of heat conduction
• Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds. [A contribution to the Gallavotti-Cohen chaotic hypothesis.]
• On convergence of linear response formulae in some piecewise hyperbolic maps