On the susceptibility function of piecewise expanding interval maps

Abstract

We study the susceptibility function Psi(z) associated to the perturbation f_t=f+tX of a piecewise expanding interval map f. The analysis is based on a spectral description of transfer operators. It gives in particular sufficient conditions which guarantee that Psi(z) is holomorphic in a disc of larger than one. Although Psi(1) is the formal derivative of the SRB measure of f_t with respect to t, we present examples satisfying our conditions so that the SRB measure is not Lipschitz.*We propose a new version of Ruelle's conjectures.* In v2, we corrected a few minor mistakes and added Conjectures A-B and Remark 4.5. In v3, we corrected the perturbation (X(f(x)) instead of X(x)), in particular in the examples from Section 6. As a consequence, Psi(z) has a pole at z=1 for these examples.

To appear Comm. Math. Phys

References in corpus (4)

• Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem
• Differentiating the absolutely continuous invariant measure of an interval map f with respect to f
• Analyticity of the Susceptibility Function for Unimodal Markovian Maps of the Interval
• On the hyperbolicity of the Feigenbaum fixed point

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