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Spectra of expanding maps on Besov spaces
Linear response for Dirac observables of Anosov diffeomorphisms
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 |
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'.
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. |



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