Linear response, or else

TL;DR

This paper reviews recent advances and open problems in the theory of linear response for chaotic dynamical systems, focusing on differentiability of invariant measures under parameter changes, including bifurcations.

Contribution

It summarizes recent results and highlights open problems in the linear response theory for chaotic systems with parameter-dependent dynamics.

Findings

Linear response holds under certain conditions for chaotic systems.

Differentiability of invariant measures can be characterized in specific settings.

Open problems remain in understanding linear response near bifurcations.

Abstract

Consider a smooth one-parameter family t -> f_t of dynamical systems f_t, with |t|<epsilon. Assume that for all t (or for many t close to t=0) the map f_t admits a unique SRB invariant probability measure m_t. We say that linear response} holds if t -> m_t is differentiable at t=0 (possibly in the sense of Whitney), and if its derivative can be expressed as a function of f_0, m_0, and d_t f_t|_(t=0). The goal of this note is to present to a general mathematical audience recent results and open problems in the theory of linear response for chaotic dynamical systems, possibly with bifurcations.