Linear response for intermittent maps

V. Baladi, M. Todd

Published 2015-08-11Version 1

We consider the one parameter family a-> T_a (a in [0,1]) of Pomeau-Manneville type interval maps introduced by Liverani-Saussol-Vaienti, with their associated absolutely continuous invariant probability measure m_a. For a in (0,1), Sarig and Gou\"ezel proved that the system mixes only polynomially (in particular, there is no spectral gap). We show that for any bounded observable g the map sending a to the average of g with respect to m_a is differentiable on [0,1/2), and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula is obtained for slowly mixing dynamics. Our argument shows how cone techniques can be used to achieve linear response.