NDLI: Linear Response for Intermittent Maps

Content Provider	Springer Nature Link
Author	Baladi, Viviane Todd, Mike
Copyright Year	2016
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.
Starting Page	857
Ending Page	874
Page Count	18
File Format	PDF
ISSN	00103616
Journal	Communications in Mathematical Physics
Volume Number	347
Issue Number	3
e-ISSN	14320916
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-02-22
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Theoretical, Mathematical and Computational Physics Mathematical Physics Quantum Physics Complex Systems Classical and Quantum Gravitation, Relativity Theory
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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