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Differentiating potential functions of SRB measures on hyperbolic attractors

Published online by Cambridge University Press:  10 June 2011

MIAOHUA JIANG
MIAOHUA JIANG*
Affiliation:
Department of Mathematics, Wake Forest University, Winston Salem, NC 27109, USA (email: jiangm@wfu.edu)
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Abstract

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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 4 , August 2012 , pp. 1350 - 1369
Copyright
Copyright © Cambridge University Press 2011

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