Singularities of the susceptibility of a Sinai-Ruelle-Bowen measure in the presence of stable-unstable tangencies.

Let ρ be a Sinai-Ruelle-Bowen (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, non-rigorous, 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) thus has a chance to be defined.