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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.

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##### 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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