Linear response, or else

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.

ICM Seoul 2014 talk

References in corpus (10)

• A review of linear response theory for general differentiable dynamical systems
• A simple framework to justify linear response theory
• Linear response formula for piecewise expanding unimodal maps
• On the susceptibility function of piecewise expanding interval maps
• Differentiating the absolutely continuous invariant measure of an interval map f with respect to f
• Summability implies Collet-Eckmann almost surely
• Smooth deformations of piecewise expanding unimodal maps
• Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps
• Singularities of the susceptibility of an SRB measure in the presence of stable-unstable tangencies
• Generalised continuation by means of right limits

Cited by in corpus (20)

• Linear response for intermittent maps
• Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables
• On spurious detection of linear response and misuse of the fluctuation-dissipation theorem in finite time series
• Controlling the statistical properties of expanding maps
• On the probability of finding a nonphysical solution through shadowing
• A regularity result for fixed points, with applications to linear response
• Optimal linear response for Markov Hilbert-Schmidt integral operators and stochastic dynamical systems
• Self consistent transfer operators. Invariant measures, convergence to equilibrium, linear response and control of the statistical properties
• Statistical properties of dynamics. Introduction to the functional analytic approach
• Linear response theory for diffeomorphisms with tangencies of stable and unstable manifolds. [A contribution to the Gallavotti-Cohen chaotic hypothesis.]
• Perturbation theory for the Fokker-Planck operator in chaos
• Steady states, fluctuation-dissipation theorems and homogenization for diffusions in a random environment with finite range of dependence
• On the fractional susceptibility function of piecewise expanding maps
• A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds
• Rigorous justification for the space-split sensitivity algorithm to compute linear response in Anosov systems
• Fast differentiation of hyperbolic chaos
• Linear response for spiking neuronal networks with unbounded memory
• Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Itô Diffusions
• A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics
• SRB measures for partially hyperbolic attractors of local diffeomorphisms