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December  2016, 36(12): 6657-6668. doi: 10.3934/dcds.2016089

Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm

Wael Bahsoun 1, and  Benoît Saussol 2,

1. 

Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU
2. 

Université de Brest, Laboratoire de Mathématiques de Bretagne Atlantique, CNRS UMR 6205, Brest, France

Received  January 2016 Revised  July 2016 Published  October 2016

We provide a general framework to study differentiability of SRB measures for one dimensional non-uniformly expanding maps. Our technique is based on inducing the non-uniformly expanding system to a uniformly expanding one, and on showing how the linear response formula of the non-uniformly expanding system is inherited from the linear response formula of the induced one. We apply this general technique to interval maps with a neutral fixed point (Pomeau-Manneville maps) to prove differentiability of the corresponding SRB measure. Our work covers systems that admit a finite SRB measure and it also covers systems that admit an infinite SRB measure. In particular, we obtain a linear response formula for both finite and infinite SRB measures. To the best of our knowledge, this is the first work that contains a linear response result for infinite measure preserving systems.
Keywords: intermittent maps., Linear response.
Mathematics Subject Classification: Primary: 37A05, 37E0.
Citation: Wael Bahsoun, Benoît Saussol. Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6657-6668. doi: 10.3934/dcds.2016089
References:
[1]

W. Bahsoun, C. Bose and Y. Duan, Rigorous Pointwise approximations for invariant densities of nonuniformly expanding maps, Ergodic Theory and Dynamical Systems, 35 (2015), 1028-1044. doi: 10.1017/etds.2013.91.  Google Scholar

[2]

W. Bahsoun, S. Galatolo, I. Nisoli and X. Niu, A Rigorous Computational Approach to Linear Response,, Available at , ().   Google Scholar

[3]

W. Bahsoun and S. Vaienti, Metastability of certain intermittent maps, Nonlinearity, 25 (2012), 107-124. doi: 10.1088/0951-7715/25/1/107.  Google Scholar

[4]

V. Baladi, On the susceptibility function of piecewise expanding interval maps, Comm. Math. Phy., 275 (2007), 839-859. doi: 10.1007/s00220-007-0320-5.  Google Scholar

[5]

V. Baladi, Linear response, or else,, Available at , ().   Google Scholar

[6]

V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677-711. doi: 10.1088/0951-7715/21/4/003.  Google Scholar

[7]

V. Baladi and M. Todd, Linear response for intermittent maps, Comm. Math. Phy., 347, (2016), 857-874. Google Scholar

[8]

O. Butterley and C. Liverani, Smooth Anosov flows: Correlation spectra and stability, J. Mod. Dyn., 1 (2007), 301-322. doi: 10.3934/jmd.2007.1.301.  Google Scholar

[9]

D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155 (2004), 389-449. doi: 10.1007/s00222-003-0324-5.  Google Scholar

[10]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217. doi: 10.1017/S0143385705000374.  Google Scholar

[11]

A. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math., 98 (1989), 581-597. doi: 10.1007/BF01393838.  Google Scholar

[12]

A. Korepanov, Linear response for intermittent maps with summable and nonsummable decay of correlations, Nonlinearity, 29 (2016), 1735-1754, Available at http://arxiv.org/abs/1508.06571. doi: 10.1088/0951-7715/29/6/1735.  Google Scholar

[13]

C. Liverani, Invariant measures and their properties. a functional analytic point of view, Dynamical systems., Part II, 185-237, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003.  Google Scholar

[14]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic theory Dynam. System, 19 (1999), 671-685. doi: 10.1017/S0143385799133856.  Google Scholar

[15]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), 189-197. doi: 10.1007/BF01197757.  Google Scholar

[16]

D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241. doi: 10.1007/s002200050134.  Google Scholar

[17]

L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.  Google Scholar

show all references

References:
[1]

W. Bahsoun, C. Bose and Y. Duan, Rigorous Pointwise approximations for invariant densities of nonuniformly expanding maps, Ergodic Theory and Dynamical Systems, 35 (2015), 1028-1044. doi: 10.1017/etds.2013.91.  Google Scholar

[2]

W. Bahsoun, S. Galatolo, I. Nisoli and X. Niu, A Rigorous Computational Approach to Linear Response,, Available at , ().   Google Scholar

[3]

W. Bahsoun and S. Vaienti, Metastability of certain intermittent maps, Nonlinearity, 25 (2012), 107-124. doi: 10.1088/0951-7715/25/1/107.  Google Scholar

[4]

V. Baladi, On the susceptibility function of piecewise expanding interval maps, Comm. Math. Phy., 275 (2007), 839-859. doi: 10.1007/s00220-007-0320-5.  Google Scholar

[5]

V. Baladi, Linear response, or else,, Available at , ().   Google Scholar

[6]

V. Baladi and D. Smania, Linear response formula for piecewise expanding unimodal maps, Nonlinearity, 21 (2008), 677-711. doi: 10.1088/0951-7715/21/4/003.  Google Scholar

[7]

V. Baladi and M. Todd, Linear response for intermittent maps, Comm. Math. Phy., 347, (2016), 857-874. Google Scholar

[8]

O. Butterley and C. Liverani, Smooth Anosov flows: Correlation spectra and stability, J. Mod. Dyn., 1 (2007), 301-322. doi: 10.3934/jmd.2007.1.301.  Google Scholar

[9]

D. Dolgopyat, On differentiability of SRB states for partially hyperbolic systems, Invent. Math., 155 (2004), 389-449. doi: 10.1007/s00222-003-0324-5.  Google Scholar

[10]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217. doi: 10.1017/S0143385705000374.  Google Scholar

[11]

A. Katok, G. Knieper, M. Pollicott and H. Weiss, Differentiability and analyticity of topological entropy for Anosov and geodesic flows, Invent. Math., 98 (1989), 581-597. doi: 10.1007/BF01393838.  Google Scholar

[12]

A. Korepanov, Linear response for intermittent maps with summable and nonsummable decay of correlations, Nonlinearity, 29 (2016), 1735-1754, Available at http://arxiv.org/abs/1508.06571. doi: 10.1088/0951-7715/29/6/1735.  Google Scholar

[13]

C. Liverani, Invariant measures and their properties. a functional analytic point of view, Dynamical systems., Part II, 185-237, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003.  Google Scholar

[14]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergodic theory Dynam. System, 19 (1999), 671-685. doi: 10.1017/S0143385799133856.  Google Scholar

[15]

Y. Pomeau and P. Manneville, Intermittent transition to turbulence in dissipative dynamical systems, Comm. Math. Phys., 74 (1980), 189-197. doi: 10.1007/BF01197757.  Google Scholar

[16]

D. Ruelle, Differentiation of SRB states, Comm. Math. Phys., 187 (1997), 227-241. doi: 10.1007/s002200050134.  Google Scholar

[17]

L.-S. Young, Recurrence times and rates of mixing, Israel J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.  Google Scholar

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