Divergence formulas for recursively sampling perturbations 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. Then, for physical measures of hyperbolic chaotic systems, we derive an equivariant divergence formula for the unstable derivative of transfer operators. This formula and hence the derivative of physical measures can be sampled by only $2u+1$ recursive relations on one orbit, where $u$ is the unstable dimension. The numerical implementation of this formula in \cite{arxiv:2111.07692} is neither cursed by dimensionality nor the butterfly effect.