We show that the derivative of the (measure) transfer operator with respect to the parameter of the map is a divergence. Then, for physical measures of discrete-time hyperbolic chaotic systems, we derive an equivariant divergence formula for the unstable perturbation of transfer operators along unstable manifolds. This formula and hence the linear response, the parameter-derivative of physical measures, can be sampled by recursively computing only $2u$ many vectors on one orbit, where $u$ is the unstable dimension. The numerical implementation of this formula in \cite{far} is neither cursed by dimensionality nor the sensitive dependence on initial conditions.

翻译：我们证明（测度）传递算子对映射参数的导数是一个散度。然后，对于离散时间双曲混沌系统的物理测度，我们推导出一个等变散度公式，用于沿着不稳定流形对传递算子进行不稳定扰动。该公式及由此产生的线性响应（物理测度对参数的导数）可通过在一条轨道上递归仅计算$2u$个向量来采样，其中$u$为不稳定维数。该公式在文献\cite{far}中的数值实现既不受维数灾难影响，也不受对初始条件敏感依赖性的制约。