On the Susceptibility Function of Piecewise Expanding Interval Maps

Abstract

We study the susceptibility function

$$\Psi(z)=\sum_{n=0}^\infty \int z^n X (y) \rho_0(y) \frac {\partial}{\partial y} \varphi (f^n (y))\, dy$$

associated to the perturbation $f_t\,=\,f\,+\,tX\,\circ\,f$ of a piecewise expanding interval map f, and to an observable φ. Ψ(1) is the formal derivative (at t  =  0) of the average ${\mathcal{R}}(t)\,=\,\int\varphi\rho_t\,dx$ of φ with respect to the SRB measure of f _t . Our analysis is based on a spectral description of transfer operators. It gives in particular sufficient conditions on f, X, and φ which guarantee that Ψ(z) is holomorphic in a disc of larger than one, or which ensure that a number may be associated to the possibly divergent series Ψ(1) . We present examples of f, X, and φ so that ${\mathcal{R}}(t)$ is not Lipschitz at 0, and we propose a new version of Ruelle’s conjecture.