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The 2019 Michael Brin Prize in Dynamical Systems
The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces
Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA |
We review recent advances in the spectral approach to studying statistical properties of dynamical systems highlighting, in particular, the role played by Sébastien Gouëzel.
References:
[1] |
J. Aaronson and D. Terhesiu,
Local limit theorems for suspended semiflows, Discrete Contin. Dyn. Syst., 40 (2020), 6575-6609.
doi: 10.3934/dcds.2020294. |
[2] |
J. F. Alves, W. Bahsoun and M. Ruziboev, Almost sure rates of mixing for partially hyperbolic attractors, preprint, arXiv: 1904.12844. Google Scholar |
[3] |
M. F. Atiyah and R. Bott,
A Lefschetz fixed point formula for elliptic complexes. I., Ann. of Math. (2), 86 (1967), 374-407.
doi: 10.2307/1970694. |
[4] |
A. Avila and S. Gouëzel,
Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, Ann. of Math. (2), 178 (2013), 385-442.
doi: 10.4007/annals.2013.178.2.1. |
[5] |
A. Avila, S. Gouëzel and M. Tsujii,
Smoothness of solenoidal attractors, Discrete Contin. Dyn. Syst., 15 (2006), 21-35.
doi: 10.3934/dcds.2006.15.21. |
[6] |
A. Avila, S. Gouëzel and J.-C. Yoccoz,
Exponential mixing for the Teichmüller flow, Publ. Math Inst. Hautes Études Sci., 104 (2006), 143-211.
doi: 10.1007/s10240-006-0001-5. |
[7] |
V. Baladi, Dynamical zeta functions and dynamical determinants for hyperbolic maps. A functional approach, in Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics, 68, Springer, Cham, 2018.
doi: 10.1007/978-3-319-77661-3. |
[8] |
V. Baladi, There are no deviations for the ergodic averages of the Giulietti-Liverani horocycle flows on the two-torus, preprint, to appear in Ergodic Theory Dynam. Systems. Google Scholar |
[9] |
V. Baladi and M. F. Demers, Thermodynamic formalism for dispersing billiards, preprint, arXiv: 2009.10936. Google Scholar |
[10] |
V. Baladi, M. F. Demers and C. Liverani,
Exponential decay of correlations for finite horizon Sinai billiard flows, Invent. Math., 211 (2018), 39-177.
doi: 10.1007/s00222-017-0745-1. |
[11] |
V. Baladi and S. Gouëzel,
Good Banach spaces for piecewise hyperbolic maps via interpolation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1453-1481.
doi: 10.1016/j.anihpc.2009.01.001. |
[12] |
V. Baladi and S. Gouëzel,
Banach spaces for piecewise cone-hyperbolic maps, J. Mod. Dyn., 4 (2010), 91-137.
doi: 10.3934/jmd.2010.4.91. |
[13] |
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. |
[14] |
P. Bálint, N. Chernov and D. Dolgopyat,
Limit theorems for dispersing billiards with cusps, Comm. Math. Phys., 308 (2011), 479-510.
doi: 10.1007/s00220-011-1342-6. |
[15] |
P. Bálint and S. Gouëzel,
Limit theorems in the stadium billiard, Comm. Math. Phys., 263 (2006), 461-512.
doi: 10.1007/s00220-005-1511-6. |
[16] |
P. M. Bleher,
Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon, J. Stat. Phys., 66 (1992), 315-373.
doi: 10.1007/BF01060071. |
[17] |
O. Butterley and L. D. Simonelli,
Parabolic flows renormalized by partially hyperbolic maps, Boll. Unione Mat. Ital., 13 (2020), 341-360.
doi: 10.1007/s40574-020-00235-8. |
[18] |
R. T. Bortolotti,
Physical measures for certain partially hyperbolic attractors on 3-manifolds, Ergodic Theory Dynam. Systems, 39 (2019), 74-104.
doi: 10.1017/etds.2017.24. |
[19] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, , 2$^nd$ edition, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008. |
[20] |
M. Blank, G. Keller and C. Liverani,
Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.
doi: 10.1088/0951-7715/15/6/309. |
[21] |
R. Castorrini and C. Liverani, Quantitative statistical properties of two-dimensional partially hyperbolic systems, preprint, arXiv: 2007.05602. Google Scholar |
[22] |
M. Cekić, S. Dyatlov, B. Kuster and G. P. Paternain, The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds, preprint, arXiv: 2009.08558. Google Scholar |
[23] |
J.-R. Chazottes and S. Gouëzel,
On almost-sure versions of classical limit theorems for dynamical systems, Probab. Theory Related Fields, 138 (2007), 195-234.
doi: 10.1007/s00440-006-0021-6. |
[24] |
P. Collet and S. Isola,
On the essential spectrum of the transfer operator for expanding Markov maps, Comm. Math. Phys., 139 (1991), 551-557.
doi: 10.1007/BF02101879. |
[25] |
J. Dedecker, S. Gouëzel and F. Merlevède, Large and moderate deviations for bounded functions of slowly mixing Markov chains, Stoch. Dyn., 18 (2018), 1850017, 38 pp.
doi: 10.1142/S021949371850017X. |
[26] |
M. F. Demers and C. Liverani,
Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc., 360 (2008), 4777-4814.
doi: 10.1090/S0002-9947-08-04464-4. |
[27] |
M. F. Demers and H.-K. Zhang,
Spectral analysis of the transfer operator for the Lorentz gas, J. Mod. Dyn., 5 (2011), 665-709.
doi: 10.3934/jmd.2011.5.665. |
[28] |
M. F. Demers and H.-K. Zhang,
A functional analytic approach to perturbations of the Lorentz gas, Comm. Math. Phys., 324 (2013), 767-830.
doi: 10.1007/s00220-013-1820-0. |
[29] |
M. F. Demers and H.-K. Zhang,
Spectral analysis of hyperbolic systems with singularities, Nonlinearity, 27 (2014), 379-433.
doi: 10.1088/0951-7715/27/3/379. |
[30] |
W. Doeblin and R. Fortet,
Sur des chaînes à liaisons complètes, Bull. Soc. Math. France, 65 (1937), 132-148.
|
[31] |
D. Dolgopyat and P. Nándori,
On mixing and the local central limit theorem for hyperbolic flows, Ergodic Theory Dynam. Systems, 40 (2020), 142-174.
doi: 10.1017/etds.2018.29. |
[32] |
S. Dyatlov and M. Zworski, Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. Éc. Norm. Supér (4), 49 (2016), 543–577.
doi: 10.24033/asens.2290. |
[33] |
S. Dyatlov and M. Zworski,
Ruelle zeta function at zero for surfaces, Invent. Math., 210 (2017), 211-229.
doi: 10.1007/s00222-017-0727-3. |
[34] |
F. Faure, S. Gouëzel and E. Lanneau, Ruelle spectrum of linear pseudo-Anosov maps, J. Éc. Polytech. Math., 6 (2019), 811–877.
doi: 10.5802/jep.107. |
[35] |
F. Faure and M. Tsujii, Band structure of the Ruelle spectrum of contact Anosov flows, C. R. Math. Acad. Sci. Paris, 351 (2013), 9–10, 385–391.
doi: 10.1016/j.crma.2013.04.022. |
[36] |
F. Faure and M. Tsujii,
Resonances for geodesic flows on negatively curved manifolds, Proceedings of the International Congress of Mathematicians – Seoul 2014, 3 (2014), 683-697.
|
[37] |
F. Faure and M. Tsujii,
The semiclassical zeta function for geodesic flows on negatively curved manifolds, Invent. Math., 208 (2017), 851-998.
doi: 10.1007/s00222-016-0701-5. |
[38] |
F. Faure and M. Tsujii, Fractal Weyl law for the Ruelle spectrum of Anosov flows, preprint, arXiv: 1706.09307. Google Scholar |
[39] |
D. Fried,
Meromorphic zeta functions for analytic flows, Comm. Math. Phys., 174 (1995), 161-190.
doi: 10.1007/BF02099469. |
[40] |
G. Forni, Ruelle resonances from cohomological equations, arXiv: 2007.03116. Google Scholar |
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G. Forni, On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces, arXiv: 2007.03144. Google Scholar |
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P. Giulietti and C. Liverani,
Parabolic dynamics and anisotropic Banach spaces, J. Eur. Math. Soc. (JEMS), 21 (2019), 2793-2858.
doi: 10.4171/JEMS/892. |
[43] |
P. Giulietti, C. Liverani and M. Pollicott,
Anosov flows and dynamical zeta functions, Ann. of Math. (2), 178 (2013), 687-773.
doi: 10.4007/annals.2013.178.2.6. |
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S. Gouëzel,
Sharp polynomial estimates for the decay of correlations, Israel J. Math., 139 (2004), 29-65.
doi: 10.1007/BF02787541. |
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S. Gouëzel,
Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields, 128 (2004), 82-122.
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S. Gouëzel,
Berry-Esseen theorem and local limit theorem for non uniformly expanding maps, Ann. Inst. H. Poincaré Probab. Statist., 41 (2005), 997-1024.
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Almost sure invariance principle for dynamical systems by spectral methods, Ann. Probab., 38 (2010), 1639-1671.
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Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Systems, 26 (2006), 189-217.
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Compact locally maximal hyperbolic sets for smooth maps: Fine statistical properties, J. Differential Geom., 79 (2008), 433-477.
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Mixing for some non-uniformly hyperbolic systems, Ann. Henri Poincaré, 17 (2016), 179-226.
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Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, 23 (2010), 1495-1545.
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show all references
References:
[1] |
J. Aaronson and D. Terhesiu,
Local limit theorems for suspended semiflows, Discrete Contin. Dyn. Syst., 40 (2020), 6575-6609.
doi: 10.3934/dcds.2020294. |
[2] |
J. F. Alves, W. Bahsoun and M. Ruziboev, Almost sure rates of mixing for partially hyperbolic attractors, preprint, arXiv: 1904.12844. Google Scholar |
[3] |
M. F. Atiyah and R. Bott,
A Lefschetz fixed point formula for elliptic complexes. I., Ann. of Math. (2), 86 (1967), 374-407.
doi: 10.2307/1970694. |
[4] |
A. Avila and S. Gouëzel,
Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmüller flow, Ann. of Math. (2), 178 (2013), 385-442.
doi: 10.4007/annals.2013.178.2.1. |
[5] |
A. Avila, S. Gouëzel and M. Tsujii,
Smoothness of solenoidal attractors, Discrete Contin. Dyn. Syst., 15 (2006), 21-35.
doi: 10.3934/dcds.2006.15.21. |
[6] |
A. Avila, S. Gouëzel and J.-C. Yoccoz,
Exponential mixing for the Teichmüller flow, Publ. Math Inst. Hautes Études Sci., 104 (2006), 143-211.
doi: 10.1007/s10240-006-0001-5. |
[7] |
V. Baladi, Dynamical zeta functions and dynamical determinants for hyperbolic maps. A functional approach, in Ergebnisse der Mathematik und ihrer Grenzgebiete, A Series of Modern Surveys in Mathematics, 68, Springer, Cham, 2018.
doi: 10.1007/978-3-319-77661-3. |
[8] |
V. Baladi, There are no deviations for the ergodic averages of the Giulietti-Liverani horocycle flows on the two-torus, preprint, to appear in Ergodic Theory Dynam. Systems. Google Scholar |
[9] |
V. Baladi and M. F. Demers, Thermodynamic formalism for dispersing billiards, preprint, arXiv: 2009.10936. Google Scholar |
[10] |
V. Baladi, M. F. Demers and C. Liverani,
Exponential decay of correlations for finite horizon Sinai billiard flows, Invent. Math., 211 (2018), 39-177.
doi: 10.1007/s00222-017-0745-1. |
[11] |
V. Baladi and S. Gouëzel,
Good Banach spaces for piecewise hyperbolic maps via interpolation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1453-1481.
doi: 10.1016/j.anihpc.2009.01.001. |
[12] |
V. Baladi and S. Gouëzel,
Banach spaces for piecewise cone-hyperbolic maps, J. Mod. Dyn., 4 (2010), 91-137.
doi: 10.3934/jmd.2010.4.91. |
[13] |
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. |
[14] |
P. Bálint, N. Chernov and D. Dolgopyat,
Limit theorems for dispersing billiards with cusps, Comm. Math. Phys., 308 (2011), 479-510.
doi: 10.1007/s00220-011-1342-6. |
[15] |
P. Bálint and S. Gouëzel,
Limit theorems in the stadium billiard, Comm. Math. Phys., 263 (2006), 461-512.
doi: 10.1007/s00220-005-1511-6. |
[16] |
P. M. Bleher,
Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon, J. Stat. Phys., 66 (1992), 315-373.
doi: 10.1007/BF01060071. |
[17] |
O. Butterley and L. D. Simonelli,
Parabolic flows renormalized by partially hyperbolic maps, Boll. Unione Mat. Ital., 13 (2020), 341-360.
doi: 10.1007/s40574-020-00235-8. |
[18] |
R. T. Bortolotti,
Physical measures for certain partially hyperbolic attractors on 3-manifolds, Ergodic Theory Dynam. Systems, 39 (2019), 74-104.
doi: 10.1017/etds.2017.24. |
[19] |
R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, , 2$^nd$ edition, Lecture Notes in Mathematics, 470, Springer-Verlag, Berlin, 2008. |
[20] |
M. Blank, G. Keller and C. Liverani,
Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, 15 (2002), 1905-1973.
doi: 10.1088/0951-7715/15/6/309. |
[21] |
R. Castorrini and C. Liverani, Quantitative statistical properties of two-dimensional partially hyperbolic systems, preprint, arXiv: 2007.05602. Google Scholar |
[22] |
M. Cekić, S. Dyatlov, B. Kuster and G. P. Paternain, The Ruelle zeta function at zero for nearly hyperbolic 3-manifolds, preprint, arXiv: 2009.08558. Google Scholar |
[23] |
J.-R. Chazottes and S. Gouëzel,
On almost-sure versions of classical limit theorems for dynamical systems, Probab. Theory Related Fields, 138 (2007), 195-234.
doi: 10.1007/s00440-006-0021-6. |
[24] |
P. Collet and S. Isola,
On the essential spectrum of the transfer operator for expanding Markov maps, Comm. Math. Phys., 139 (1991), 551-557.
doi: 10.1007/BF02101879. |
[25] |
J. Dedecker, S. Gouëzel and F. Merlevède, Large and moderate deviations for bounded functions of slowly mixing Markov chains, Stoch. Dyn., 18 (2018), 1850017, 38 pp.
doi: 10.1142/S021949371850017X. |
[26] |
M. F. Demers and C. Liverani,
Stability of statistical properties in two-dimensional piecewise hyperbolic maps, Trans. Amer. Math. Soc., 360 (2008), 4777-4814.
doi: 10.1090/S0002-9947-08-04464-4. |
[27] |
M. F. Demers and H.-K. Zhang,
Spectral analysis of the transfer operator for the Lorentz gas, J. Mod. Dyn., 5 (2011), 665-709.
doi: 10.3934/jmd.2011.5.665. |
[28] |
M. F. Demers and H.-K. Zhang,
A functional analytic approach to perturbations of the Lorentz gas, Comm. Math. Phys., 324 (2013), 767-830.
doi: 10.1007/s00220-013-1820-0. |
[29] |
M. F. Demers and H.-K. Zhang,
Spectral analysis of hyperbolic systems with singularities, Nonlinearity, 27 (2014), 379-433.
doi: 10.1088/0951-7715/27/3/379. |
[30] |
W. Doeblin and R. Fortet,
Sur des chaînes à liaisons complètes, Bull. Soc. Math. France, 65 (1937), 132-148.
|
[31] |
D. Dolgopyat and P. Nándori,
On mixing and the local central limit theorem for hyperbolic flows, Ergodic Theory Dynam. Systems, 40 (2020), 142-174.
doi: 10.1017/etds.2018.29. |
[32] |
S. Dyatlov and M. Zworski, Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. Éc. Norm. Supér (4), 49 (2016), 543–577.
doi: 10.24033/asens.2290. |
[33] |
S. Dyatlov and M. Zworski,
Ruelle zeta function at zero for surfaces, Invent. Math., 210 (2017), 211-229.
doi: 10.1007/s00222-017-0727-3. |
[34] |
F. Faure, S. Gouëzel and E. Lanneau, Ruelle spectrum of linear pseudo-Anosov maps, J. Éc. Polytech. Math., 6 (2019), 811–877.
doi: 10.5802/jep.107. |
[35] |
F. Faure and M. Tsujii, Band structure of the Ruelle spectrum of contact Anosov flows, C. R. Math. Acad. Sci. Paris, 351 (2013), 9–10, 385–391.
doi: 10.1016/j.crma.2013.04.022. |
[36] |
F. Faure and M. Tsujii,
Resonances for geodesic flows on negatively curved manifolds, Proceedings of the International Congress of Mathematicians – Seoul 2014, 3 (2014), 683-697.
|
[37] |
F. Faure and M. Tsujii,
The semiclassical zeta function for geodesic flows on negatively curved manifolds, Invent. Math., 208 (2017), 851-998.
doi: 10.1007/s00222-016-0701-5. |
[38] |
F. Faure and M. Tsujii, Fractal Weyl law for the Ruelle spectrum of Anosov flows, preprint, arXiv: 1706.09307. Google Scholar |
[39] |
D. Fried,
Meromorphic zeta functions for analytic flows, Comm. Math. Phys., 174 (1995), 161-190.
doi: 10.1007/BF02099469. |
[40] |
G. Forni, Ruelle resonances from cohomological equations, arXiv: 2007.03116. Google Scholar |
[41] |
G. Forni, On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces, arXiv: 2007.03144. Google Scholar |
[42] |
P. Giulietti and C. Liverani,
Parabolic dynamics and anisotropic Banach spaces, J. Eur. Math. Soc. (JEMS), 21 (2019), 2793-2858.
doi: 10.4171/JEMS/892. |
[43] |
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