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April  2019, 39(4): 1799-1819. doi: 10.3934/dcds.2019078

Linear response for Dirac observables of Anosov diffeomorphisms

Matthieu Porte 1,2,

1. 

École normale supérieure, Département de mathématiques et applications, Paris, 75005, France
2. 

Current affiliation, AgroParisTech, Département de sciences économiques, sociales et de gestion, Paris, 75015, France

Received  October 2017 Revised  September 2018 Published  January 2019

Fund Project: This work was conducted in 2017 as a MSc thesis at UPMC, under the direction of V. Baladi (CNRS/IMJ-PRG), whom the author thanks for her guidance and her useful remarks at all stages of this work. The author thanks P-A. Guihneuf (UMPC/IMJ-PRG) for his helpful comments. The author is also grateful to the reviewers for their insights on this paper throughout the drafting process.

We consider a $ \mathcal{C}^3 $ family $ t\mapsto f_t $ of $ \mathcal{C}^4 $ Anosov diffeomorphisms on a compact Riemannian manifold $ M $. Denoting by $ \rho_t $ the SRB measure of $ f_t $, we prove that the map $ t\mapsto\int \theta d\rho_t $ is differentiable if $ \theta $ is of the form $ \theta(x) = h(x)\delta(g(x)-a) $, with $ \delta $ the Dirac distribution, $ g:M\rightarrow \mathbb{R} $ a $ \mathcal{C}^4 $ function, $ h:M\rightarrow\mathbb{C} $ a $ \mathcal{C}^3 $ function and $ a $ a regular value of $ g $. We also require a transversality condition, namely that the intersection of the support of $ h $ with the level set $ \{g(x) = a\} $ is foliated by 'admissible stable leaves'.

Keywords: Linear response, discontinuous observables, Anosov diffeomorphisms, hyperbolic dynamics, transfer operators, anisotropic Banach spaces.
Mathematics Subject Classification: Primary: 37C40; Secondary: 37C30, 37D20.
Citation: Matthieu Porte. Linear response for Dirac observables of Anosov diffeomorphisms. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1799-1819. doi: 10.3934/dcds.2019078
References:
[1]

V. Baladi, Linear response, or else, Proceedings of the ICM - Seoul, 2014, 525–545.

[2]

V. Baladi, Positive Transfer Operators and Decay of Correlations, vol. 16 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.

[3]

V. Baladi, The quest for the ultimate anisotropic Banach space, J. Stat. Phys., 166 (2017), 525-557.  doi: 10.1007/s10955-016-1663-0.

[4]

V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Springer Ergebnisse Vol 68, 2018. doi: 10.1007/978-3-319-77661-3.

[5]

V. Baladi, T. Kuna and V. Lucarini, Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables, Nonlinearity, 30 (2017), 1204-1220, URL http://stacks.iop.org/0951-7715/30/i=3/a=1204, Corrigendum Nonlinearity, 30 (2017), C4-C6. doi: 10.1088/1361-6544/aa5b13.

[6]

V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble), 57 (2007), 127-154.  doi: 10.5802/aif.2253.

[7]

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

[8]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634.  doi: 10.2307/2160348.

[9]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

[10]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995, With a supplementary chapter by Katok and Leonardo Mendoza. doi: 10.1017/CBO9780511809187.

[11]

V. Lucarini, F. Ragone and F. Lunkeit, Predicting climate change using response theory: global averages and spatial patterns, J. Stat. Phys., 166 (2017), 1036-1064.  doi: 10.1007/s10955-016-1506-z.

[12]

J. N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 Indag. Math., 30 (1968), 479-483. 

[13]

R. D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478, URL http://projecteuclid.org/euclid.dmj/1077379127. doi: 10.1215/S0012-7094-70-03759-2.

[14]

D. Ruelle, A measure associated with Axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810.

[15]

D. Ruelle, A review of linear response theory for general differentiable dynamical systems, Nonlinearity, 22 (2009), 855-870.  doi: 10.1088/0951-7715/22/4/009.

[16]

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

[17]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.

show all references

References:
[1]

V. Baladi, Linear response, or else, Proceedings of the ICM - Seoul, 2014, 525–545.

[2]

V. Baladi, Positive Transfer Operators and Decay of Correlations, vol. 16 of Advanced Series in Nonlinear Dynamics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812813633.

[3]

V. Baladi, The quest for the ultimate anisotropic Banach space, J. Stat. Phys., 166 (2017), 525-557.  doi: 10.1007/s10955-016-1663-0.

[4]

V. Baladi, Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps, Springer Ergebnisse Vol 68, 2018. doi: 10.1007/978-3-319-77661-3.

[5]

V. Baladi, T. Kuna and V. Lucarini, Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables, Nonlinearity, 30 (2017), 1204-1220, URL http://stacks.iop.org/0951-7715/30/i=3/a=1204, Corrigendum Nonlinearity, 30 (2017), C4-C6. doi: 10.1088/1361-6544/aa5b13.

[6]

V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier (Grenoble), 57 (2007), 127-154.  doi: 10.5802/aif.2253.

[7]

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

[8]

H. Hennion, Sur un théorème spectral et son application aux noyaux lipchitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634.  doi: 10.2307/2160348.

[9]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition.

[10]

A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, vol. 54 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1995, With a supplementary chapter by Katok and Leonardo Mendoza. doi: 10.1017/CBO9780511809187.

[11]

V. Lucarini, F. Ragone and F. Lunkeit, Predicting climate change using response theory: global averages and spatial patterns, J. Stat. Phys., 166 (2017), 1036-1064.  doi: 10.1007/s10955-016-1506-z.

[12]

J. N. Mather, Characterization of Anosov diffeomorphisms, Nederl. Akad. Wetensch. Proc. Ser. A 71 Indag. Math., 30 (1968), 479-483. 

[13]

R. D. Nussbaum, The radius of the essential spectrum, Duke Math. J., 37 (1970), 473-478, URL http://projecteuclid.org/euclid.dmj/1077379127. doi: 10.1215/S0012-7094-70-03759-2.

[14]

D. Ruelle, A measure associated with Axiom-A attractors, Amer. J. Math., 98 (1976), 619-654.  doi: 10.2307/2373810.

[15]

D. Ruelle, A review of linear response theory for general differentiable dynamical systems, Nonlinearity, 22 (2009), 855-870.  doi: 10.1088/0951-7715/22/4/009.

[16]

P. Walters, An Introduction to Ergodic Theory, vol. 79 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1982.

[17]

L.-S. Young, What are SRB measures, and which dynamical systems have them?, J. Statist. Phys., 108 (2002), 733-754.  doi: 10.1023/A:1019762724717.

Figure 1.  Behavior of the support of a Dirac observable on the stable manifold under the reverse dynamics. The two axes through the origin are the unstable and stable directions. The parallel lines are level sets for the function g(u, s) = u with the line passing through the origin being the level set {g = 0}. In the left side picture, the shaded area is the support of h. The right side picture is the image of the left side picture by f−1. Both pictures are in $\mathbb{R}^2$, each of the grid squares corresponds to a fundamental domain of the torus.
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Figure 2.  Behavior of the support of a Dirac observable with g(x, y) = y − 0.4x under the reverse dynamics. The stable and unstable directions are the same as befor. Parallel lines are level sets for g. The line through the origin is the level set {g = 0}. The shaded area is the support of h. The right side picture is the image of the left side one by f−1.
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Figure 3.  Behavior of the support of a Dirac observable with g(x, y) = x2 + y2 under the reverse dynamics. In the left side picture, circles are level sets of g and the middle circle is the level set {g = 0.42}. The shaded area is the support of h. Note that it excludes the region where the middle circle is tangent to the unstable direction. The right side picture is the image of the left side one by f−1.
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