Application of hyperbolic dynamics to physics: Some problems and conjectures

The long time behavior of smooth dynamical systems is, in good cases, given by an SRB measure $\rho$. The measure $\rho$ is expected often to have very discontinuous dependence on parameters of the dynamical system. This is a very unsatisfactory situation for physical applications, where one would like to differentiate $\rho$ with respect to parameters. Here we propose a solution to this difficulty, based on analytic continuation in a frequency variable $\omega$.