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The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces
Three problems solved by Sébastien Gouëzel
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).
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. |
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. |
[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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