On differentiability of SRB states for partially hyperbolic systems

• 成果类型: Article
• 署名作者: Dolgopyat, D
• 署名单位: University System of Maryland; University of Maryland College Park
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-003-0324-5
• 发表日期: 2004
• 页码: 389-449
• 关键词: invariant-measures; stable ergodicity; DYNAMICAL-SYSTEMS; anosov; entropy; sequence; families; map

摘要

Consider a one parameter family of diffeomorphisms f(epsilon) such that f(0) is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let nu(epsilon) be any u-Gibbs state for f(epsilon). We prove (Theorem 1) that for any C-infinity function A the map epsilon-->nu(epsilon)(A) is differentiable at epsilon=0. This implies (Corollary 2.2) that the difference of Birkhoff averages of the perturbed and unperturbed systems is proportional to epsilon. We apply this result (Corollary 3.3) to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique SRB measure with good statistical properties.