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.

Minor typos corrected. To appear in Comm. Math. Phys

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• Regularity of characteristic exponents and linear response for transfer operator cocycles
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• A spectral approach to quenched linear and higher-order response for partially hyperbolic dynamics