Abstract
We consider the one parameter family α↦T_{α} (α∈[0,1)) of PomeauManneville type interval maps T_{α}(x)=x(1+2^{α}x^{α}) for x∈[0,1/2) and T_{α}(x)=2x−1 for x∈[1/2,1], with the associated absolutely continuous invariant probability measure μ_{α}. For α∈(0,1), Sarig and Gouëzel proved that the system mixes only polynomially with rate n^{1−1/α} (in particular, there is no spectral gap). We show that for any ψ∈L^{q}, the map α→∫^{1}_{0}ψdμ_{α} 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 α≥1/2 we need the n^{−1/α} decorrelation obtained by Gouëzel under additional conditions.
Original language  English 

Pages (fromto)  857874 
Number of pages  18 
Journal  Communications in Mathematical Physics 
Volume  347 
Issue number  3 
Early online date  22 Feb 2016 
DOIs  
Publication status  Published  Nov 2016 
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Mike Todd
 School of Mathematics and Statistics  Deputy Head of School
 Pure Mathematics  Professor
Person: Academic