American Institute of Mathematical Sciences

Advanced
  2020, 16: 351-371. doi: 10.3934/jmd.2020014

The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces (Brin Prize article)

Dmitry Dolgopyat

Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA

Published  December 2020

We review recent advances in the spectral approach to studying statistical properties of dynamical systems highlighting, in particular, the role played by Sébastien Gouëzel.

Keywords: Hyperbolicity, limit theorems, quasi compactness, Lasota-Yorke inequality, Pollicott-Ruelle resonances, induced map.
Mathematics Subject Classification: Primary:37A50, 37C30;Secondary:37D20, 37D25, 37D30, 60F05.
Citation: Dmitry Dolgopyat. The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces. Journal of Modern Dynamics, 2020, 16: 351-371. doi: 10.3934/jmd.2020014
References:
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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.

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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.

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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.

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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.

[9]

V. Baladi and M. F. Demers, Thermodynamic formalism for dispersing billiards, preprint, arXiv: 2009.10936.

[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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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W. Doeblin and R. Fortet, Sur des chaînes à liaisons complètes, Bull. Soc. Math. France, 65 (1937), 132-148. 

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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.

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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.  doi: 10.1007/s00440-003-0300-4.

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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.  doi: 10.1016/j.anihpb.2004.09.002.

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show all references

References:
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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.

[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.

[9]

V. Baladi and M. F. Demers, Thermodynamic formalism for dispersing billiards, preprint, arXiv: 2009.10936.

[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.

[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.

[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.

[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.

[41]

G. Forni, On the equidistribution of unstable curves for pseudo-Anosov diffeomorphisms of compact surfaces, arXiv: 2007.03144.

[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]

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.

[44]

S. Gouëzel, Sharp polynomial estimates for the decay of correlations, Israel J. Math., 139 (2004), 29-65.  doi: 10.1007/BF02787541.

[45]

S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields, 128 (2004), 82-122.  doi: 10.1007/s00440-003-0300-4.

[46]

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Figure 1.  Semidispersing billiards
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