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）传递算子关于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，我们推导出一个沿不稳定流形扰动不稳定传递算子的等变散度公式。该公式及其线性响应（物理测度的参数导数）可以通过在一条轨道上仅递归计算$2u$个向量进行采样，其中$u$是不稳定维数。\cite{far}中该公式的数值实现既不受维数灾难的困扰，也不受初值敏感依赖性的影响。