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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Published in | Ergodic theory and dynamical systems Vol. 32; no. 4; pp. 1350 - 1369 |
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Main Author | |
Format | Journal Article |
Language | English |
Published |
Cambridge, UK
Cambridge University Press
01.08.2012
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Subjects | |
Online Access | Get full text |
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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. |
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Bibliography: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
ISSN: | 0143-3857 1469-4417 |
DOI: | 10.1017/S0143385711000241 |