On differentiability of SRB states for partially hyperbolic systems

@article{Dolgopyat2004OnDO,
  title={On differentiability of SRB states for partially hyperbolic systems},
  author={Dmitry Dolgopyat},
  journal={Inventiones mathematicae},
  year={2004},
  volume={155},
  pages={389-449}
}
  • D. Dolgopyat
  • Published 1 February 2004
  • Mathematics
  • Inventiones mathematicae
Consider a one parameter family of diffeomorphisms fε such that f0 is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let νε be any u-Gibbs state for fε. We prove (Theorem 1) that for any C∞ function A the map ε→νε(A) is differentiable at ε=0. This implies (Corollary 2.2) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to ε. We apply this result (Corollary 3.3) to show that a generic… 

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