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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: viviane.baladi@ens.fr)
DANIEL SMANIA
Affiliation:
Departamento de Matemática, ICMC-USP, Caixa Postal 668, São Carlos-SP, CEP 13560-970, São Carlos-SP, Brazil (email: smania@icmc.usp.br)
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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

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