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.