| 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. |
