Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds—a contribution to the Gallavotti–Cohen chaotic hypothesis

Abstract

This note presents a non-rigorous study of the linear response for an SRB (or ‘natural physical’) measure ρ of a diffeomorphism f in the presence of tangencies of the stable and unstable manifolds of ρ. We propose that generically, if ρ has no zero Lyapunov exponent, if its stable dimension is sufficiently large (greater than 1/2 or perhaps 3/2) and if it is exponentially mixing in a suitable sense, then the following formal expression for the first derivative of with respect to f along X is convergent:This suggests that an SRB measure may exist for small perturbations of f, with weak differentiability.