Linear Response for Intermittent Maps

Abstract

We consider the one parameter family ${\alpha \mapsto T_{\alpha}}$ (${\alpha \in [0,1)}$) of Pomeau-Manneville type interval maps ${T_{\alpha}(x) = x(1+2^{\alpha} x^{\alpha})}$ for ${x \in [0,1/2)}$ and ${T_{\alpha}(x)=2x-1}$ for ${x \in [1/2, 1]}$, with the associated absolutely continuous invariant probability measure ${\mu_{\alpha}}$. For ${\alpha \in (0,1)}$, Sarig and Gouëzel proved that the system mixes only polynomially with rate ${n^{1-1/{\alpha}}}$ (in particular, there is no spectral gap). We show that for any ${\psi \in L^{q}}$, the map ${\alpha \to \int_0^{1} \psi\, d \mu_{\alpha}}$ is differentiable on ${[0,1-1/q)}$, and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For ${\alpha \ge 1/2}$ we need the ${n^{-1/{\alpha}}}$ decorrelation obtained by Gouëzel under additional conditions.