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September  2013, 33(9): 4239-4269. doi: 10.3934/dcds.2013.33.4239

Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts

Jean-Pierre Conze 1, and  Y. Guivarc'h 2,

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

Received  May 2011 Revised  July 2011 Published  March 2013

Let $(X, \cal B, \nu)$ be a probability space and let $\Gamma$ be a countable group of $\nu$-preserving invertible maps of $X$ into itself. To a probability measure $\mu$ on $\Gamma$ corresponds a random walk on $X$ with Markov operator $P$ given by $P\psi(x) = \sum_{a} \psi(ax) \, \mu(a)$. We consider various examples of ergodic $\Gamma$-actions and random walks and their extensions by a vector space: groups of automorphisms or affine transformations on compact nilmanifolds, random walks in random scenery on non amenable groups, translations on homogeneous spaces of simple Lie groups, random walks on motion groups. A powerful tool in this study is the spectral gap property for the operator $P$ when it holds. We use it to obtain limit theorems, recurrence/transience property and ergodicity for random walks on non compact extensions of the corresponding dynamical systems.
Keywords: spectral gap, recurrence, local limit theorem, non compact extension of dynamical system, random walk, random scenery., Nilmanifold.
Mathematics Subject Classification: Primary: 37A30, 37A40, 28D05, 22D40, 60F0.
Citation: Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
References:
[1]

J. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[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.  Google Scholar

[3]

B. Bekka, P. de la Harpe and A. Valette, "Kazhdan's Property (T)," New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008.  Google Scholar

[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.  Google Scholar

[5]

B. Bekka and Y. Guivarc'h, On the spectral theory of groups of affine transformations on compact nilmanifolds,, , ().   Google Scholar

[6]

L. Breiman, "Probability," Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1968.  Google Scholar

[7]

B. M. Brown, Martingale central limit theorems, Ann. Math. Statist., 42 (1971), 59-66.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[10]

J. Dixmier and W. G. Lister, Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc., 8 (1957), 155-158.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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M. I. Gordin and B. A. Lifšic, Central limit theorem for stationary Markov processes, (Russian) Dokl. Akad. Nauk SSSR, 239 (1978), 766-767.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

V. Kaimanovich, The Poisson boundary of covering Markov operators, Israel J. Math., 89 (1995), 77-134. doi: 10.1007/BF02808195.  Google Scholar

[21]

S. A. Kalikow, $T,T^{-1}$ transformation is not loosely Bernoulli, Ann. of Math. (2), 115 (1982), 393-409. doi: 10.2307/1971397.  Google Scholar

[22]

D. A. Kazhdan, Uniform distribution on a plane, (Russian) Trudy Moskov. Mat. Ob., 14 (1965), 299-305.  Google Scholar

[23]

H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc., 92 (1959), 336-354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. A. Margulis, "Discrete Subgroups of Semisimple Lie Groups," Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991.  Google Scholar

[28]

W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math., 91 (1969), 757-771.  Google Scholar

[29]

W. Parry, Dynamical systems on nilmanifolds, Bull. London Math. Soc., 2 (1970), 37-40.  Google Scholar

[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.  Google Scholar

[31]

K. Schmidt, "Lectures on Cocycles of Ergodic Transformations Groups," Lect. Notes in Math., Vol. 1, Mac Millan Co. of India, Ltd., Delhi, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

Ye. Shalom, Explicit Kazhdan constants for representations of semisimple and arithmetic groups, Ann. Inst. Fourier (Grenoble), 50 (2000), 833-863.  Google Scholar

[35]

J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

P. P. Varjú, Random walks in Euclidean spaces,, , ().   Google Scholar

[39]

R. Zimmer, "Ergodic Theory and Semisimple Groups," Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984.  Google Scholar

show all references

References:
[1]

J. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997.  Google Scholar

[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.  Google Scholar

[3]

B. Bekka, P. de la Harpe and A. Valette, "Kazhdan's Property (T)," New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008.  Google Scholar

[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.  Google Scholar

[5]

B. Bekka and Y. Guivarc'h, On the spectral theory of groups of affine transformations on compact nilmanifolds,, , ().   Google Scholar

[6]

L. Breiman, "Probability," Addison-Wesley Publishing Company, Reading, Mass.-London-Don Mills, Ont., 1968.  Google Scholar

[7]

B. M. Brown, Martingale central limit theorems, Ann. Math. Statist., 42 (1971), 59-66.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

J. Dixmier and W. G. Lister, Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc., 8 (1957), 155-158.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. I. Gordin and B. A. Lifšic, Central limit theorem for stationary Markov processes, (Russian) Dokl. Akad. Nauk SSSR, 239 (1978), 766-767.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

V. Kaimanovich, The Poisson boundary of covering Markov operators, Israel J. Math., 89 (1995), 77-134. doi: 10.1007/BF02808195.  Google Scholar

[21]

S. A. Kalikow, $T,T^{-1}$ transformation is not loosely Bernoulli, Ann. of Math. (2), 115 (1982), 393-409. doi: 10.2307/1971397.  Google Scholar

[22]

D. A. Kazhdan, Uniform distribution on a plane, (Russian) Trudy Moskov. Mat. Ob., 14 (1965), 299-305.  Google Scholar

[23]

H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc., 92 (1959), 336-354.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. A. Margulis, "Discrete Subgroups of Semisimple Lie Groups," Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991.  Google Scholar

[28]

W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math., 91 (1969), 757-771.  Google Scholar

[29]

W. Parry, Dynamical systems on nilmanifolds, Bull. London Math. Soc., 2 (1970), 37-40.  Google Scholar

[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.  Google Scholar

[31]

K. Schmidt, "Lectures on Cocycles of Ergodic Transformations Groups," Lect. Notes in Math., Vol. 1, Mac Millan Co. of India, Ltd., Delhi, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

Ye. Shalom, Explicit Kazhdan constants for representations of semisimple and arithmetic groups, Ann. Inst. Fourier (Grenoble), 50 (2000), 833-863.  Google Scholar

[35]

J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250-270.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

P. P. Varjú, Random walks in Euclidean spaces,, , ().   Google Scholar

[39]

R. Zimmer, "Ergodic Theory and Semisimple Groups," Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984.  Google Scholar

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