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Differentiation of SRB states for hyperbolic flows

Published online by Cambridge University Press:  01 April 2008

DAVID RUELLE
DAVID RUELLE*
Affiliation:
Mathematics Department, Rutgers University, and IHES, 91440 Bures sur Yvette, France (email: ruelle@ihes.fr)
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Abstract

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Let the ${\cal C}^3$ vector field ${\cal X}+aX$ on M define a flow (fat) with an Axiom A attractor Λa depending continuously on a∈(−ϵ,ϵ). Let ρa be the SRB measure on Λa for (fat). If $A\in {\cal C}^2(M)$, then $a\mapsto \rho _a(A)$ is ${\cal C}^1$ on (−ϵ,ϵ) and dρa(A)/da is the limit when ω→0 with Im ω>0 of \[ \int _0^\infty e^{i\omega t}\,dt \int \rho _a (d x)\,X(x)\cdot \nabla _x(A\circ f_a^t). \]

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 2: William Parry Memorial Volume , April 2008, pp. 613 - 631
Copyright
Copyright © Cambridge University Press 2008

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