Three problems solved by Sébastien Gouëzel

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The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces

2020, 16: 373-387. doi: 10.3934/jmd.2020015

Three problems solved by Sébastien Gouëzel (Brin Prize article)

François Ledrappier

Sorbonne Université and CNRS, UMR 8001, LPSM, Case courrier 158, 4 Place Jussieu, F-75252, Paris Cedex 05, France

Published December 2020

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We present three results of Sébastien Gouëzel's: the local limit theorem for random walks on hyperbolic groups, a multiplicative ergodic theorem for non-expansive mappings (joint work with Anders Karlsson), and the description of the essential spectrum of the Laplacian on $ SL(2,{\mathbb R}) $ orbits in the moduli space (joint work with Artur Avila).

Keywords: Limit theorems, spectral gap.

Mathematics Subject Classification: Primary:37D40;37A17;Secondary:37A25;37A30;37A50.

Citation: François Ledrappier. Three problems solved by Sébastien Gouëzel. Journal of Modern Dynamics, 2020, 16: 373-387. doi: 10.3934/jmd.2020015

References:

[1]	J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398. doi: 10.1007/s002220000057.
[2]	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.
[3]	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.
[4]	A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2), 125 (1987), 495-536. doi: 10.2307/1971409.
[5]	A. Ancona, Théorie du potentiel sur les graphes et les variétés, École d'été de Probabilités de Saint-Flour XVIII–-1988, Lecture Notes in Math., 1427, Springer, Berlin, 1990, 5–112. doi: 10.1007/BFb0103041.
[6]	P. Bougerol, Théorème central limite local sur certains groupes de Lie, Ann. Sci. École Norm. Sup. (4), 14 (1981), 403-432. doi: 10.24033/asens.1412.
[7]	D. Dolgopyat, The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces, J. Mod. Dyn., 16 (2020), 351-371. doi: 10.3934/jmd.2020014.
[8]	A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225.
[9]	A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL (2, $\Bbb R$) action on moduli space, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 95-324. doi: 10.1007/s10240-018-0099-2.
[10]	G. Forni, On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations, J. Mod. Dyn., 6 (2012), 139-182. doi: 10.3934/jmd.2012.6.139.
[11]	S. Gouëzel, Local limit theorem for symmetric random walks in Gromov-hyperbolic groups, J. Amer. Math. Soc., 27 (2014), 893-928. doi: 10.1090/S0894-0347-2014-00788-8.
[12]	S. Gouëzel, Martin boundary of random walks with unbounded jumps in hyperbolic groups, Ann. Probab., 43 (2015), 2374-2404. doi: 10.1214/14-AOP938.
[13]	S. Gouëzel and A. Karlsson, Subadditive and multiplicative ergodic theorems, J. Euro. Math. Soc. (JEMS), 22 (2020), 1893-1915. doi: 10.4171/JEMS/958.
[14]	S. Gouëzel and S. P. Lalley, Random walks on co-compact Fuchsian groups, Ann. Sci. Éc. Norm. Supér. (4), 46 (2013), 129-173. doi: 10.24033/asens.2186.
[15]	S. Gouëzel and V. Shchur, Corrigendum: A corrected quantitative version of the Morse Lemma, J. Funct. Anal., 277 (2019), 1258-1268. doi: 10.1016/j.jfa.2019.02.021.
[16]	P. Gerl and W. Woess, Local limits and harmonic functions for nonisotropic random walks on free groups, Probab. Theory Relat. Fields, 71 (1986), 341-355. doi: 10.1007/BF01000210.
[17]	H. Hennion, Sur un théorème spectral et son application aux noyaux lipschitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634. doi: 10.2307/2160348.
[18]	M. Izumi, S. Neshveyev and R. Okayasu, The ratio set of the harmonic measure of a random walk on a hyperbolic group, Israel J. Math., 163 (2008), 285-316. doi: 10.1007/s11856-008-0013-6.
[19]	V. A. Kaimanovich, Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups, J. Soviet Math., 47 (1989), 2387-2398. doi: 10.1007/BF01840421.
[20]	V. A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 361-393.
[21]	J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B, 30 (1968), 499-510. doi: 10.1111/j.2517-6161.1968.tb00749.x.
[22]	A. Karlsson and G. A. Margulis, A multiplicative ergodic theorem and nonpositively curved spaces, Comm. Math. Phys., 208 (1999), 107-123. doi: 10.1007/s002200050750.
[23]	S. P. Lalley, Finite range random walk on free groups and homogeneous trees, Ann. Probab., 21 (1993), 2087-2130. doi: 10.1214/aop/1176989012.
[24]	C. Liverani, On Contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312. doi: 10.4007/annals.2004.159.1275.
[25]	V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obsc., 19 (1968), 179-210.
[26]	J. Parkinson, Isotropic random walks on affine buildings, Ann. Inst. Fourier (Grenoble), 57 (2007), 379-419. doi: 10.5802/aif.2262.
[27]	W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187–188 (1990), 268 pp.
[28]	M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic. Theory Dynam. Systems, 7 (1987), 267-288. doi: 10.1017/S0143385700004004.

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References:

[1]	J. F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math., 140 (2000), 351-398. doi: 10.1007/s002220000057.
[2]	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.
[3]	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.
[4]	A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2), 125 (1987), 495-536. doi: 10.2307/1971409.
[5]	A. Ancona, Théorie du potentiel sur les graphes et les variétés, École d'été de Probabilités de Saint-Flour XVIII–-1988, Lecture Notes in Math., 1427, Springer, Berlin, 1990, 5–112. doi: 10.1007/BFb0103041.
[6]	P. Bougerol, Théorème central limite local sur certains groupes de Lie, Ann. Sci. École Norm. Sup. (4), 14 (1981), 403-432. doi: 10.24033/asens.1412.
[7]	D. Dolgopyat, The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces, J. Mod. Dyn., 16 (2020), 351-371. doi: 10.3934/jmd.2020014.
[8]	A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478. doi: 10.1017/S0143385701001225.
[9]	A. Eskin and M. Mirzakhani, Invariant and stationary measures for the SL (2, $\Bbb R$) action on moduli space, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 95-324. doi: 10.1007/s10240-018-0099-2.
[10]	G. Forni, On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations, J. Mod. Dyn., 6 (2012), 139-182. doi: 10.3934/jmd.2012.6.139.
[11]	S. Gouëzel, Local limit theorem for symmetric random walks in Gromov-hyperbolic groups, J. Amer. Math. Soc., 27 (2014), 893-928. doi: 10.1090/S0894-0347-2014-00788-8.
[12]	S. Gouëzel, Martin boundary of random walks with unbounded jumps in hyperbolic groups, Ann. Probab., 43 (2015), 2374-2404. doi: 10.1214/14-AOP938.
[13]	S. Gouëzel and A. Karlsson, Subadditive and multiplicative ergodic theorems, J. Euro. Math. Soc. (JEMS), 22 (2020), 1893-1915. doi: 10.4171/JEMS/958.
[14]	S. Gouëzel and S. P. Lalley, Random walks on co-compact Fuchsian groups, Ann. Sci. Éc. Norm. Supér. (4), 46 (2013), 129-173. doi: 10.24033/asens.2186.
[15]	S. Gouëzel and V. Shchur, Corrigendum: A corrected quantitative version of the Morse Lemma, J. Funct. Anal., 277 (2019), 1258-1268. doi: 10.1016/j.jfa.2019.02.021.
[16]	P. Gerl and W. Woess, Local limits and harmonic functions for nonisotropic random walks on free groups, Probab. Theory Relat. Fields, 71 (1986), 341-355. doi: 10.1007/BF01000210.
[17]	H. Hennion, Sur un théorème spectral et son application aux noyaux lipschitziens, Proc. Amer. Math. Soc., 118 (1993), 627-634. doi: 10.2307/2160348.
[18]	M. Izumi, S. Neshveyev and R. Okayasu, The ratio set of the harmonic measure of a random walk on a hyperbolic group, Israel J. Math., 163 (2008), 285-316. doi: 10.1007/s11856-008-0013-6.
[19]	V. A. Kaimanovich, Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semisimple Lie groups, J. Soviet Math., 47 (1989), 2387-2398. doi: 10.1007/BF01840421.
[20]	V. A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré Phys. Théor., 53 (1990), 361-393.
[21]	J. F. C. Kingman, The ergodic theory of subadditive stochastic processes, J. Roy. Statist. Soc. Ser. B, 30 (1968), 499-510. doi: 10.1111/j.2517-6161.1968.tb00749.x.
[22]	A. Karlsson and G. A. Margulis, A multiplicative ergodic theorem and nonpositively curved spaces, Comm. Math. Phys., 208 (1999), 107-123. doi: 10.1007/s002200050750.
[23]	S. P. Lalley, Finite range random walk on free groups and homogeneous trees, Ann. Probab., 21 (1993), 2087-2130. doi: 10.1214/aop/1176989012.
[24]	C. Liverani, On Contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312. doi: 10.4007/annals.2004.159.1275.
[25]	V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obsc., 19 (1968), 179-210.
[26]	J. Parkinson, Isotropic random walks on affine buildings, Ann. Inst. Fourier (Grenoble), 57 (2007), 379-419. doi: 10.5802/aif.2262.
[27]	W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187–188 (1990), 268 pp.
[28]	M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic. Theory Dynam. Systems, 7 (1987), 267-288. doi: 10.1017/S0143385700004004.

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