Paper detail

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

Let $ρ$ be an SRB (or "physical"), measure for the discrete time evolution given by a map $f$, and let $ρ(A)$ denote the expectation value of a smooth function $A$. If $f$ depends on a parameter, the derivative $δρ(A)$ of $ρ(A)$ with respect to the parameter is formally given by the value of the so-called susceptibility function $Ψ(z)$ at $z=1$. When $f$ is a uniformly hyperbolic diffeomorphism, it has been proved that the power series $Ψ(z)$ has a radius of convergence $r(Ψ)>1$, and that $δρ(A)=Ψ(1)$, but it is known that $r(Ψ)<1$ 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 $(f,ρ)$. The present paper gives a crude, nonrigorous, analysis of this situation in terms of the Hausdorff dimension $d$ of $ρ$ in the stable direction. We find that the tangencies produce singularities of $Ψ(z)$ 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 $Ψ(1)$ makes sense, and the derivative $δρ(A)=Ψ(1)$ has thus a chance to be defined