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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

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.
Citation: Wael Bahsoun, Benoît Saussol. Linear response in the intermittent family: Differentiation in a weighted $C^0$-norm. Discrete and 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.

[2]

W. Bahsoun, S. Galatolo, I. Nisoli and X. Niu, A Rigorous Computational Approach to Linear Response, Available at http://arxiv.org/abs/1506.08661

[3]

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

[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.

[5]

V. Baladi, Linear response, or else, Available at http://arxiv.org/pdf/1408.2937v1.pdf

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

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.

[2]

W. Bahsoun, S. Galatolo, I. Nisoli and X. Niu, A Rigorous Computational Approach to Linear Response, Available at http://arxiv.org/abs/1506.08661

[3]

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

[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.

[5]

V. Baladi, Linear response, or else, Available at http://arxiv.org/pdf/1408.2937v1.pdf

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[16]

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

[17]

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

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