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

Wael Bahsoun; Benoît  Saussol

doi:10.3934/dcds.2016089

Article Contents

2016, Volume 36, Issue 12: 6657-6668. Doi: 10.3934/dcds.2016089

This issue Previous Article Haldane linearisation done right: Solving the nonlinear recombination equation the easy way Next Article Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings

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

Received: January 2016

Revised: July 2016

Published: May 2017

Abstract / Introduction Related Papers Cited by

Abstract

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:

Linear response,
intermittent maps.

Mathematics Subject Classification: Primary: 37A05, 37E05.

Citation:

Related Papers

Cited by

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.