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

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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

Abstract
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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 and 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.
[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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