  We formulate a linear phase and frequency response theory for hyperbolic
flows, which generalizes phase response theory for autonomous limit cycle
oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing
conjecture, stating the existence of a perturbed trajectory shadowing every
unperturbed trajectory on the system attractor for any small enough
perturbation of arbitrary duration and a corresponding unique time isomorphism,
which we identify as phase, such that phase shifts between the unperturbed
trajectory and its perturbed shadow are well defined. The phase sensitivity
function is the solution of an adjoint linear equation and can be used to
estimate the average change of phase velocity to small time dependent or
independent perturbations. These changes of frequency are experimentally
accessible giving a convenient way to define and measure phase response curves
for chaotic oscillators. The shadowing trajectory and the phase can be
constructed explicitly in the tangent space of an unperturbed trajectory using
co-variant Lyapunov vectors. It can also be used to identify the limits of the
regime of linear response.
