Recursive divergence formulas for perturbing unstable transfer operators and physical measures

TL;DR

This paper introduces recursive divergence formulas for perturbing unstable transfer operators in chaotic systems, enabling efficient computation of physical measure derivatives without suffering from high dimensionality or initial condition sensitivity.

Contribution

It derives a novel equivariant divergence formula for unstable perturbations, facilitating linear response calculations with a recursive, dimension-independent method.

Findings

The derivative of the transfer operator is expressed as a divergence.

The formula allows sampling with only 2u vectors, where u is the unstable dimension.

Numerical implementation avoids curse of dimensionality and sensitivity issues.

Abstract

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.