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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: [email protected])
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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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References

[1]Anosov, D.. Geodesic Flows on Closed Riemann Manifolds with Negative Curvature (Proceedings of the Stecklov Institute of Mathematics, 90). American Mathematical Society, Providence RI, 1969 (English Translation).Google Scholar
[2]Baladi, V.. Linear response despite critical points. Nonlinearity 21(6) (2008), T81–T90.CrossRefGoogle Scholar
[3]Baladi, V. and Gouëzel, S.. Good Banach spaces for piecewise hyperbolic maps via interpolation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(4) (2009), 1453–1481.CrossRefGoogle Scholar
[4]Baladi, V. and Smania, D.. Alternative proofs of linear response for piecewise expanding unimodal maps. Ergod. Th. & Dynam. Sys. 30(1) (2010), 1–20.CrossRefGoogle Scholar
[5]Baladi, V. and Smania, D.. Analyticity of the SRB measure for holomorphic families of quadratic-like Collet–Eckmann maps. Proc. Amer. Math. Soc. 137(4) (2009), 1431–1437.CrossRefGoogle Scholar
[6]Barreira, L. and Pesin, Y.. Nonuniform hyperbolicity. Dynamics of Systems with Nonzero Lyapunov Exponents (Encyclopedia of Mathematics and its Applications, 115). Cambridge University Press, Cambridge, 2007.CrossRefGoogle Scholar
[7]Blank, M., Keller, G. and Liverani, C.. Ruelle–Perron–Frobenius spectrum for Anosov maps. Nonlinearity 15(6) (2001), 1905–1973.CrossRefGoogle Scholar
[8]Butterley, O. and Liverani, C.. Smooth Anosov flows: correlation spectra and stability. J. Mod. Dyn. 1(2) (2007), 301–322.CrossRefGoogle Scholar
[9]Baladi, V. and Smania, D.. Linear response formula for piecewise expanding unimodal maps. Nonlinearity 21(4) (2008), 677–711.CrossRefGoogle Scholar
[10]Baladi, V. and Tsujii, M.. Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Ann. Inst. Fourier (Grenoble) 57(1) (2007), 127–154.CrossRefGoogle Scholar
[11]Dolgopyat, D.. On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155(2) (2004), 389–449.CrossRefGoogle Scholar
[12]Gouëzel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26(1) (2006), 189–217.CrossRefGoogle Scholar
[13]Hirsch, M. W., Pugh, C. C. and Shub, M.. Invariant Manifolds (Lecture Notes in Mathematics, 583). Springer, New York, 1977.CrossRefGoogle Scholar
[14]Jiang, M. and de la Llave, R.. Smooth dependence of thermodynamic limits of SRB measures. Comm. Math. Phys. 211 (2000), 303–333.CrossRefGoogle Scholar
[15]Jiang, M. and de la Llave, R.. Linear response function for coupled hyperbolic attractors. Comm. Math. Phys. 261(2) (2006), 379–404.CrossRefGoogle Scholar
[16]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theorey of Dynamical Systems. Cambridge University Press, Cambridge, 1996.Google Scholar
[17]Pollicott, M.. Stability of mixing rates for Axiom A attractors. Nonlinearity 16 (2003), 567–578.CrossRefGoogle Scholar
[18]Ruelle, D.. Differentiation of SRB States. Comm. Math. Phys. 187 (1997), 227–241.CrossRefGoogle Scholar
[19]Ruelle, D.. Correction and complements: differentiation of SRB states. Comm. Math. Phys. 234(1) (2003), 185–190.CrossRefGoogle Scholar
[20]Ruelle, D.. Personal communication. SIAM Conf. on Applications of Dynamical Systems (Snowbird, Utah, 27–31 May 2003).Google Scholar
[21]Ruelle, D.. Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J. Stat. Phys. 95(1–2) (1999), 393–468.CrossRefGoogle Scholar
[22]Ruelle, D.. Thermodynamic Formalism (Encyclopedia of Mathematics and its Applications, 5). Addison Wesley, New York, 1978.Google Scholar
[23]Ruelle, D.. Structure and f-dependence of the A.C.I.M. for a unimodal map f of Misiurewicz type. Comm. Math. Phys. 287 (2009), 1039–1070.CrossRefGoogle Scholar
[24]Ruelle, D.. A review of linear response theory for general differentiable dynamical systems. Nonlinearity 22(4) (2009), 855–870.CrossRefGoogle Scholar
[25]Ruelle, D.. Differentiation of SRB states for hyperbolic flows. Ergod. Th. & Dynam. Sys. 28(2) (2008), 613–631.CrossRefGoogle Scholar
[26]Schmeling, J.. Hölder continuity of the holonomy maps for hyperbolic basic sets. II. Math. Nachr. 170 (1994), 211–225.CrossRefGoogle Scholar
[27]Sinai, I. G.. Topics in Ergodic Theory. Princeton University Press, Princeton, NJ, 1994.CrossRefGoogle Scholar