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

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 $\mathrm{\Psi }(z)$ at $z=1$. When $f$ is a uniformly hyperbolic diffeomorphism, it has been proved that the power series $\mathrm{\Psi }(z)$ has a radius of convergence $r(\mathrm{\Psi })>1$, and that $\delta \rho (A)=\mathrm{\Psi }(1)$, but it is known that $r(\mathrm{\Psi })<1$ 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 $\mathrm{\Psi }(z)$ for $\mathrm{\mid }z\mathrm{\mid }<1$ if $d<1\mathrm{/}2$, but only for $\mathrm{\mid }z\mathrm{\mid }>1$ if $d>1\mathrm{/}2$. In particular, if $d>1\mathrm{/}2$ we may hope that $\mathrm{\Psi }(1)$ makes sense, and the derivative $\delta \rho (A)=\mathrm{\Psi }(1)$ has thus a chance to be defined