Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds. [A contribution to the Gallavotti-Cohen chaotic hypothesis.]

TL;DR

This paper explores the linear response of SRB measures for certain diffeomorphisms with manifold tangencies, proposing conditions under which the derivative of the measure with respect to perturbations converges, indicating potential measure stability.

Contribution

It introduces a formal expression for the derivative of SRB measures in systems with stable-unstable tangencies, extending linear response theory to more complex dynamical scenarios.

Findings

Formal expression for the derivative of SRB measures converges under certain conditions.

Suggests SRB measures may exist with weak differentiability for small perturbations.

Provides a non-rigorous but insightful analysis of measure stability in complex systems.

Abstract

This note presents a non-rigorous study of the linear response for an SRB (or `natural physical') measure $\rho$ of a diffeomorphism $f$ in the presence of tangencies of the stable and unstable manifolds of $\rho$. We propose that generically, if $\rho$ has no zero Lyapunov exponent, if its stable dimension is sufficiently large (greater than 1/2 or perhaps 3/2) and if it is exponentially mixing in a suitable sense, then the following formal expression for the first derivative of $\rho(\phi)$ with respect to $f$ along $X$ is convergent: $$ \Psi(z)=\sum_{n=0}^\infty z^n\int\rho(dx)\,X(x)\cdot\nabla_x(\phi\circ f^n)\qquad{\rm for}\qquad z=1 $$ This suggests that an SRB measure may exist for small perturbations of $f$, with weak differentiability.