Ergodic Theory and Dynamical Systems

Table of Contents - 01 February 2010 - Volume 30, Issue 01  

Research Article

Alternative proofs of linear response for piecewise expanding unimodal maps

VIVIANE BALADIa1 and DANIEL SMANIAa2

a1 D.M.A., UMR 8553, École Normale Supérieure, 75005 Paris, France (email: viviane.baladi@ens.fr)

a2 Departamento de Matemática, ICMC-USP, Caixa Postal 668, São Carlos-SP, CEP 13560-970, São Carlos-SP, Brazil (email: smania@icmc.usp.br)

Abstract

We give two new proofs that the Sinai–Ruelle–Bowen (SRB) measure txs21A6μt 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. Nonlinearity 21 (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.

(Received April 09 2008)

(Revised November 17 2008)