Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds. [A contribution to the Gallavotti-Cohen chaotic hypothesis.]

This note presents a non-rigorous study of the linear response for an SRB (or `natural physical’) measure $\rho$ of a diffeomorphism $f$ in the presence of tangencies of the stable and unstable manifolds of $\rho$. We propose that generically, if $\rho$ 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 $\rho (\varphi )$ with respect to $f$ along $X$ is convergent: $$\mathrm{\Psi }(z)=\sum _{n=0}^{\mathrm{\infty }}{z}^n\int \rho (dx)X(x)\cdot {\mathrm{\nabla }}_x(\varphi \circ f^n)\mathrm{f}\mathrm{o}\mathrm{r}z=1$$ This suggests that an SRB measure may exist for small perturbations of $f$, with weak differentiability.