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1. | IRMAR, UMR CNRS 6625, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex |
2. | IRMAR, CNRS UMR 6625, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France |
References:
[1] |
J. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. |
[2] |
J. Bourgain and A. Gamburd, Spectral gaps in $ SU(d)$, C. R. Math. Acad. Sci. Paris, 348 (2010), 609-611.
doi: 10.1016/j.crma.2010.04.024. |
[3] |
B. Bekka, P. de la Harpe and A. Valette, "Kazhdan's Property (T)," New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008. |
[4] |
B. Bekka and J.-R. Heu, Random products of automorphisms of Heisenberg nilmanifolds and Weil's representation, Ergodic Theory Dynam. Systems, 31 (2011), 1277-1286.
doi: 10.1017/S014338571000043X. |
[5] |
B. Bekka and Y. Guivarc'h, On the spectral theory of groups of affine transformations on compact nilmanifolds, arXiv:1106.2623. |
[6] |
L. Breiman, "Probability," Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1968. |
[7] |
B. M. Brown, Martingale central limit theorems, Ann. Math. Statist., 42 (1971), 59-66. |
[8] |
J.-P. Conze, Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications, Ergodic Theory and Dynam. Systems, 19 (1999), 1233-1245.
doi: 10.1017/S0143385799141701. |
[9] |
J.-P. Conze and Y. Guivarc'h, Remarques sur la distalité dans les espaces vectoriels, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 1083-1086. |
[10] |
J. Dixmier and W. G. Lister, Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc., 8 (1957), 155-158. |
[11] |
A. Furman and Ye. Shalom, Sharp ergodic theorems for group actions and strong ergodicity, Ergodic Theory Dynam. Systems, 19 (1999), 1037-1061.
doi: 10.1017/S0143385799133881. |
[12] |
A. Gamburd, D. Jakobson and P. Sarnak, Spectra of elements in the group ring of $ SU(2)$, J. Eur. Math. Soc. (JEMS), 1 (1999), 51-85.
doi: 10.1007/PL00011157. |
[13] |
M. I. Gordin and B. A. Lifšic, Central limit theorem for stationary Markov processes, (Russian) Dokl. Akad. Nauk SSSR, 239 (1978), 766-767. |
[14] |
Y. Guivarc'h, Equirartition dans les espaces homogènes, (French) in "Théorie Ergodique" (Actes Journées Ergodiques, Rennes, 1973/1974), Lecture Notes in Math., Vol. 532, Springer, Berlin, (1976), 131-142. |
[15] |
Y. Guivarc'h, Limit theorems for random walks and products of random matrices, in "Probability Measures on Groups: Recent Directions and Trends," Tata Inst. Fund. Res., Mumbai, (2006), 255-330. |
[16] |
Y. Guivarc'h and J. Hardy, Théorémes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Ann. Inst. H. Poincar Probab. Statist., 24 (1988), 73-98. |
[17] |
Y. Guivarc'h and A. N. Starkov, Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms, Ergodic Theory Dynam. Systems, 24 (2004), 767-802.
doi: 10.1017/S0143385703000440. |
[18] |
Y. Guivarc'h and C. R. E. Raja, Recurrence and ergodicity of random walks on locally compact groups and on homogeneous spaces, Ergodic Theory and Dynam. Systems, 32 (2012), 1313-1349.
doi: 10.1017/S0143385711000149. |
[19] |
V. F. R. Jones and K. Schmidt, Asymptotically invariant sequences and approximate finiteness, Amer. J. Math., 109 (1987), 91-114.
doi: 10.2307/2374553. |
[20] |
V. Kaimanovich, The Poisson boundary of covering Markov operators, Israel J. Math., 89 (1995), 77-134.
doi: 10.1007/BF02808195. |
[21] |
S. A. Kalikow, $T,T^{-1}$ transformation is not loosely Bernoulli, Ann. of Math. (2), 115 (1982), 393-409.
doi: 10.2307/1971397. |
[22] |
D. A. Kazhdan, Uniform distribution on a plane, (Russian) Trudy Moskov. Mat. Ob., 14 (1965), 299-305. |
[23] |
H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc., 92 (1959), 336-354. |
[24] |
H. Kesten and F. Spitzer, A limit theorem related to a new class of self-similar processes, Z. Wahrsch. Verw. Gebiete, 50 (1979), 5-25.
doi: 10.1007/BF00535672. |
[25] |
A. Krámli and D. Szász, Random walks with internal degrees of freedom. II. first-hitting probabilities, Z. Wahrsch. Verw. Gebiete, 68 (1984), 53-64.
doi: 10.1007/BF00535173. |
[26] |
S. Le Borgne, Examples of quasi-hyperbolic dynamical systems with slow decay of correlations, C. R. Math. Acad. Sci. Paris, 343 (2006), 125-128.
doi: 10.1016/j.crma.2006.05.010. |
[27] |
G. A. Margulis, "Discrete Subgroups of Semisimple Lie Groups," Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991. |
[28] |
W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math., 91 (1969), 757-771. |
[29] |
W. Parry, Dynamical systems on nilmanifolds, Bull. London Math. Soc., 2 (1970), 37-40. |
[30] |
C. R. E. Raja, On the existence of ergodic automorphisms in ergodic $\mathbbZ^d$-actions on compact groups, Ergodic Theory Dynam. Systems, 30 (2010), 1803-1816.
doi: 10.1017/S0143385709000728. |
[31] |
K. Schmidt, "Lectures on Cocycles of Ergodic Transformations Groups," Lect. Notes in Math., Vol. 1, Mac Millan Co. of India, Ltd., Delhi, 1977. |
[32] |
K. Schmidt, Asymptotically invariant sequences and an action of $SL(2, \mathbbZ)$ on the 2-sphere, Israel J. Math., 37 (1980), 193-208.
doi: 10.1007/BF02760961. |
[33] |
K. Schmidt, On joint recurrence, C. R. Acad. Sci. Paris S. I Math., 327 (1998), 837-842.
doi: 10.1016/S0764-4442(99)80115-3. |
[34] |
Ye. Shalom, Explicit Kazhdan constants for representations of semisimple and arithmetic groups, Ann. Inst. Fourier (Grenoble), 50 (2000), 833-863. |
[35] |
J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270. |
[36] |
K. Uchiyama, Asymptotic estimates of the Green functions and transition probabilities for Markov additive processes, Electron. J. Probab., 12 (2007), 138-180.
doi: 10.1214/EJP.v12-396. |
[37] |
Ya. B. Vorobets, On the uniform distribution of orbits of finitely generated groups and semigroups of plane isometries, (Russian) Mat. Sb., 195 (2004), 17-40; translation in Sb. Math., 195 (2004), 163-186.
doi: 10.1070/SM2004v195n02ABEH000799. |
[38] |
P. P. Varjú, Random walks in Euclidean spaces, arXiv:1205.3399. |
[39] |
R. Zimmer, "Ergodic Theory and Semisimple Groups," Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984. |
show all references
References:
[1] |
J. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. |
[2] |
J. Bourgain and A. Gamburd, Spectral gaps in $ SU(d)$, C. R. Math. Acad. Sci. Paris, 348 (2010), 609-611.
doi: 10.1016/j.crma.2010.04.024. |
[3] |
B. Bekka, P. de la Harpe and A. Valette, "Kazhdan's Property (T)," New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008. |
[4] |
B. Bekka and J.-R. Heu, Random products of automorphisms of Heisenberg nilmanifolds and Weil's representation, Ergodic Theory Dynam. Systems, 31 (2011), 1277-1286.
doi: 10.1017/S014338571000043X. |
[5] |
B. Bekka and Y. Guivarc'h, On the spectral theory of groups of affine transformations on compact nilmanifolds, arXiv:1106.2623. |
[6] |
L. Breiman, "Probability," Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1968. |
[7] |
B. M. Brown, Martingale central limit theorems, Ann. Math. Statist., 42 (1971), 59-66. |
[8] |
J.-P. Conze, Sur un critère de récurrence en dimension 2 pour les marches stationnaires, applications, Ergodic Theory and Dynam. Systems, 19 (1999), 1233-1245.
doi: 10.1017/S0143385799141701. |
[9] |
J.-P. Conze and Y. Guivarc'h, Remarques sur la distalité dans les espaces vectoriels, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 1083-1086. |
[10] |
J. Dixmier and W. G. Lister, Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc., 8 (1957), 155-158. |
[11] |
A. Furman and Ye. Shalom, Sharp ergodic theorems for group actions and strong ergodicity, Ergodic Theory Dynam. Systems, 19 (1999), 1037-1061.
doi: 10.1017/S0143385799133881. |
[12] |
A. Gamburd, D. Jakobson and P. Sarnak, Spectra of elements in the group ring of $ SU(2)$, J. Eur. Math. Soc. (JEMS), 1 (1999), 51-85.
doi: 10.1007/PL00011157. |
[13] |
M. I. Gordin and B. A. Lifšic, Central limit theorem for stationary Markov processes, (Russian) Dokl. Akad. Nauk SSSR, 239 (1978), 766-767. |
[14] |
Y. Guivarc'h, Equirartition dans les espaces homogènes, (French) in "Théorie Ergodique" (Actes Journées Ergodiques, Rennes, 1973/1974), Lecture Notes in Math., Vol. 532, Springer, Berlin, (1976), 131-142. |
[15] |
Y. Guivarc'h, Limit theorems for random walks and products of random matrices, in "Probability Measures on Groups: Recent Directions and Trends," Tata Inst. Fund. Res., Mumbai, (2006), 255-330. |
[16] |
Y. Guivarc'h and J. Hardy, Théorémes limites pour une classe de chaînes de Markov et applications aux difféomorphismes d'Anosov, Ann. Inst. H. Poincar Probab. Statist., 24 (1988), 73-98. |
[17] |
Y. Guivarc'h and A. N. Starkov, Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms, Ergodic Theory Dynam. Systems, 24 (2004), 767-802.
doi: 10.1017/S0143385703000440. |
[18] |
Y. Guivarc'h and C. R. E. Raja, Recurrence and ergodicity of random walks on locally compact groups and on homogeneous spaces, Ergodic Theory and Dynam. Systems, 32 (2012), 1313-1349.
doi: 10.1017/S0143385711000149. |
[19] |
V. F. R. Jones and K. Schmidt, Asymptotically invariant sequences and approximate finiteness, Amer. J. Math., 109 (1987), 91-114.
doi: 10.2307/2374553. |
[20] |
V. Kaimanovich, The Poisson boundary of covering Markov operators, Israel J. Math., 89 (1995), 77-134.
doi: 10.1007/BF02808195. |
[21] |
S. A. Kalikow, $T,T^{-1}$ transformation is not loosely Bernoulli, Ann. of Math. (2), 115 (1982), 393-409.
doi: 10.2307/1971397. |
[22] |
D. A. Kazhdan, Uniform distribution on a plane, (Russian) Trudy Moskov. Mat. Ob., 14 (1965), 299-305. |
[23] |
H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc., 92 (1959), 336-354. |
[24] |
H. Kesten and F. Spitzer, A limit theorem related to a new class of self-similar processes, Z. Wahrsch. Verw. Gebiete, 50 (1979), 5-25.
doi: 10.1007/BF00535672. |
[25] |
A. Krámli and D. Szász, Random walks with internal degrees of freedom. II. first-hitting probabilities, Z. Wahrsch. Verw. Gebiete, 68 (1984), 53-64.
doi: 10.1007/BF00535173. |
[26] |
S. Le Borgne, Examples of quasi-hyperbolic dynamical systems with slow decay of correlations, C. R. Math. Acad. Sci. Paris, 343 (2006), 125-128.
doi: 10.1016/j.crma.2006.05.010. |
[27] |
G. A. Margulis, "Discrete Subgroups of Semisimple Lie Groups," Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991. |
[28] |
W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math., 91 (1969), 757-771. |
[29] |
W. Parry, Dynamical systems on nilmanifolds, Bull. London Math. Soc., 2 (1970), 37-40. |
[30] |
C. R. E. Raja, On the existence of ergodic automorphisms in ergodic $\mathbbZ^d$-actions on compact groups, Ergodic Theory Dynam. Systems, 30 (2010), 1803-1816.
doi: 10.1017/S0143385709000728. |
[31] |
K. Schmidt, "Lectures on Cocycles of Ergodic Transformations Groups," Lect. Notes in Math., Vol. 1, Mac Millan Co. of India, Ltd., Delhi, 1977. |
[32] |
K. Schmidt, Asymptotically invariant sequences and an action of $SL(2, \mathbbZ)$ on the 2-sphere, Israel J. Math., 37 (1980), 193-208.
doi: 10.1007/BF02760961. |
[33] |
K. Schmidt, On joint recurrence, C. R. Acad. Sci. Paris S. I Math., 327 (1998), 837-842.
doi: 10.1016/S0764-4442(99)80115-3. |
[34] |
Ye. Shalom, Explicit Kazhdan constants for representations of semisimple and arithmetic groups, Ann. Inst. Fourier (Grenoble), 50 (2000), 833-863. |
[35] |
J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270. |
[36] |
K. Uchiyama, Asymptotic estimates of the Green functions and transition probabilities for Markov additive processes, Electron. J. Probab., 12 (2007), 138-180.
doi: 10.1214/EJP.v12-396. |
[37] |
Ya. B. Vorobets, On the uniform distribution of orbits of finitely generated groups and semigroups of plane isometries, (Russian) Mat. Sb., 195 (2004), 17-40; translation in Sb. Math., 195 (2004), 163-186.
doi: 10.1070/SM2004v195n02ABEH000799. |
[38] |
P. P. Varjú, Random walks in Euclidean spaces, arXiv:1205.3399. |
[39] |
R. Zimmer, "Ergodic Theory and Semisimple Groups," Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984. |
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