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Alternative proofs of linear response for piecewise expanding unimodal maps

Published online by Cambridge University Press:  12 March 2009

VIVIANE BALADI  and
DANIEL SMANIA
VIVIANE BALADI
Affiliation:
D.M.A., UMR 8553, École Normale Supérieure, 75005 Paris, France (email: [email protected])
DANIEL SMANIA
Affiliation:
Departamento de Matemática, ICMC-USP, Caixa Postal 668, São Carlos-SP, CEP 13560-970, São Carlos-SP, Brazil (email: [email protected])
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Abstract

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We give two new proofs that the Sinai–Ruelle–Bowen (SRB) measure t↦μt of a C2 path ft of unimodal piecewise expanding C3 maps is differentiable at 0 if ft is tangent to the topological class of f0. The arguments are more conceptual than the original proof of Baladi and Smania [Linear response formula for piecewise expanding unimodal maps. Nonlinearity21 (2008), 677–711], but require proving Hölder continuity of the infinitesimal conjugacy α (a new result, of independent interest) and using spaces of bounded p-variation. The first new proof gives differentiability of higher order of ∫ ψ dμt if ft is smooth enough and stays in the topological class of f0 and if ψ is smooth enough (a new result). In addition, this proof does not require any information on the decomposition of the SRB measure into regular and singular terms, making it potentially amenable to extensions to higher dimensions. The second new proof allows us to recover the linear response formula (i.e. the formula for the derivative at 0) obtained by Baladi and Smania, by an argument more conceptual than the ‘brute force’ cancellation mechanism used by Baladi and Smania.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 1 , February 2010 , pp. 1 - 20
Copyright
Copyright © Cambridge University Press 2009

References

[1]Baladi, V.. On the susceptibility function of piecewise expanding interval maps. Comm. Math. Phys. 275 (2007), 839–859.Google Scholar
[2]Baladi, V.. Linear response despite critical points. Nonlinearity 21 (2008), T81–T90.CrossRefGoogle Scholar
[3]Baladi, V. and Hachemi, A.. A local limit theorem with speed of convergence for euclidean algorithms and diophantine costs. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), 749–770.Google Scholar
[4]Baladi, V. and Smania, D.. Linear response formula for piecewise expanding unimodal maps. Nonlinearity 21 (2008), 677–711.CrossRefGoogle Scholar
[5]Baladi, V. and Smania, D.. Smooth deformations of piecewise expanding unimodal maps. Discrete Contin. Dyn. Syst. Ser. A 23 (2009), 685–703.Google Scholar
[6]Baladi, V. and Smania, D.. Analyticity of the SRB measure for holomorphic families of quadratic-like Collet–Eckmann maps. Proc. Amer. Math. Soc. 137 (2009), 1431–1437.Google Scholar
[7]Bruneau, M.. Variation Totale d’une Fonction (Lecture Notes in Mathematics, 413). Springer, Berlin, 1974.CrossRefGoogle Scholar
[8]Buzzi, J.. Private communication (2007).Google Scholar
[9]Gouëzel, S. and Liverani, C.. Banach spaces adapted to Anosov systems. Ergod. Th. & Dynam. Sys. 26 (2006), 189–217.Google Scholar
[10]Gouëzel, S. and Liverani, C.. Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differential Geom. 79 (2008), 433–477.CrossRefGoogle Scholar
[11]Kato, T.. Perturbation Theory for Linear Operators. Springer, Berlin, 1966.Google Scholar
[12]Katok, A., Knieper, G., Pollicott, M. and Weiss, H.. Differentiability of entropy for Anosov and geodesic flows. Bull. Amer. Math. Soc. 22 (1990), 285–293.Google Scholar
[13]Katok, A., Knieper, G., Pollicott, M. and Weiss, H.. Differentiability and analyticity of topological entropy for Anosov and geodesic flows. Invent. Math. 98 (1989), 581–597.Google Scholar
[14]Keller, G.. Stochastic stability in some chaotic dynamical systems. Monatsh. Math. 94 (1982), 313–333.Google Scholar
[15]Keller, G.. Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrsch. verw. Gebiete 69 (1985), 461–478.CrossRefGoogle Scholar
[16]Keller, G. and Liverani, C.. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa 28 (1999), 141–152.Google Scholar
[17]Keller, G. and Nowicki, T.. Spectral theory, zeta functions and the distribution of periodic points for Collet–Eckmann maps. Comm. Math. Phys. 149 (1992), 31–69.Google Scholar
[18]Ruelle, D.. Differentiation of SRB states. Comm. Math. Phys. 187 (1997), 227–241; Corrections and complements. Comm. Math. Phys. 234, (2003), 185–190.Google Scholar
[19]Ruelle, D.. Nonequilibrium statistical mechanics near equilibrium: computing higher-order terms. Nonlinearity 11 (1998), 5–18.Google Scholar
[20]Ruelle, D.. Application of hyperbolic dynamics to physics: some problems and conjectures. Bull. Amer. Math. Soc. 41 (2004), 275–278.Google Scholar
[21]Ruelle, D.. Structure and f-dependence of the A.C.I.M. for a unimodal map f of Misiurewicz type. Comm. Math. Phys. to appear.Google Scholar
[22]Wong, Sh.. Hölder continuous derivatives and ergodic theory. J. London Math. Soc. 22 (1980), 506–520.Google Scholar
[23]Young, L.-S.. What are SRB measures, and which dynamical systems have them?. J. Statist. Phys. 108 (2002), 733–754.Google Scholar