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