Differentiating potential functions of SRB measures on hyperbolic attractors

The derivation of Ruelle’s derivative formula of the SRB measure depends largely on the calculation of the derivative of the unstable Jacobian. Although Ruelle’s derivative formula is correct, the proofs in the original paper and its corrigendum are not complete. In this paper, we re-visit the diffe...

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Bibliographic Details
Published inErgodic theory and dynamical systems Vol. 32; no. 4; pp. 1350 - 1369
Main Author JIANG, MIAOHUA
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.08.2012
Subjects
Chains
Derivation
Derivatives
Differentiation
Dynamical systems
Jacobians
Manifolds
Mathematical analysis
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Summary:The derivation of Ruelle’s derivative formula of the SRB measure depends largely on the calculation of the derivative of the unstable Jacobian. Although Ruelle’s derivative formula is correct, the proofs in the original paper and its corrigendum are not complete. In this paper, we re-visit the differentiation process of the unstable Jacobian and provide a complete derivation of its derivative formula. Our approach is to extend the volume form provided by the SRB measure on local unstable manifolds to a system of Hölder continuous local Riemannian metrics on the manifold so that under this system of local metrics, the unstable Jacobian becomes differentiable with respect to the base point and its derivative with respect to the map can be obtained by the chain rule.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
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content type line 23
ISSN:0143-3857
1469-4417
DOI:10.1017/S0143385711000241