Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables. (8th February 2017)

• Record Type: Journal Article
• Authors: Baladi, Viviane; Kuna, Tobias; Lucarini, Valerio
• Is Part Of: Nonlinearity. Volume 30:Number 3(2017:Mar.)
• Page Start: 1204
• Page End: 1220
• Publication Date: 2017-02-08
• Subjects: linear response -- hyperbolic attractor -- transfer operator -- anisotropic space; 37C40 -- 37C30 -- 37D20; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aa5b13
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We consider a smooth one-parameter family t ↦ ( f t : M → M ) of diffeomorphisms with compact transitive Axiom A attractors Λ t, denoting by d ρ t the SRB measure of f t | Λ t . Our first result is that for any function θ in the Sobolev space H p r ( M ), with 1 < p < ∞ and 0 < r < 1/ p, the map t ↦ ∫ θ d ρ t is α -Hölder continuous for all α < r . This applies to θ ( x ) = h ( x ) Θ ( g ( x ) − a ) (for all α < 1 ) for h and g smooth and Θ the Heaviside function, if a is not a critical value of g . Our second result says that for any such function θ ( x ) = h ( x ) Θ ( g ( x ) − a ) so that in addition the intersection of { x | g ( x ) = a } with the support of h is foliated by 'admissible stable leaves' of f t, the map t ↦ ∫ θ d ρ t is differentiable. (We provide distributional linear response and fluctuation-dissipation formulas for the derivative.) Obtaining linear response or fractional response for such observables θ is motivated by extreme-value theory.