American Institute of Mathematical Sciences

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

Three problems solved by Sébastien Gouëzel

François Ledrappier

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2), 125 (1987), 495-536.  doi: 10.2307/1971409.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478.  doi: 10.1017/S0143385701001225.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

C. Liverani, On Contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312.  doi: 10.4007/annals.2004.159.1275.  Google Scholar

[25]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obsc., 19 (1968), 179-210.   Google Scholar

[26]

J. Parkinson, Isotropic random walks on affine buildings, Ann. Inst. Fourier (Grenoble), 57 (2007), 379-419.  doi: 10.5802/aif.2262.  Google Scholar

[27]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187–188 (1990), 268 pp.  Google Scholar

[28]

M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic. Theory Dynam. Systems, 7 (1987), 267-288.  doi: 10.1017/S0143385700004004.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2), 125 (1987), 495-536.  doi: 10.2307/1971409.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Eskin and H. Masur, Asymptotic formulas on flat surfaces, Ergodic Theory Dynam. Systems, 21 (2001), 443-478.  doi: 10.1017/S0143385701001225.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

C. Liverani, On Contact Anosov flows, Ann. of Math. (2), 159 (2004), 1275-1312.  doi: 10.4007/annals.2004.159.1275.  Google Scholar

[25]

V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obsc., 19 (1968), 179-210.   Google Scholar

[26]

J. Parkinson, Isotropic random walks on affine buildings, Ann. Inst. Fourier (Grenoble), 57 (2007), 379-419.  doi: 10.5802/aif.2262.  Google Scholar

[27]

W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187–188 (1990), 268 pp.  Google Scholar

[28]

M. Ratner, The rate of mixing for geodesic and horocycle flows, Ergodic. Theory Dynam. Systems, 7 (1987), 267-288.  doi: 10.1017/S0143385700004004.  Google Scholar

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