Divergence formulas for sampling derivatives of transfer operators on an orbit

For discrete time systems, we show that the derivative of the (measure) transfer operator with respect to the system parameters is a divergence. For singular physical measures, which are limits of an orbit, we show that we typically only need the transfer operator to handle the unstable derivatives. Then we derive a divergence formula for the unstable derivative of transfer operators, which has no exploding intermediate quantities. This formula and hence the derivative of physical measures can be sampled by a few recursive relations on an orbit.