Effective decay of multiple correlations in semidirect product actions

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2016, 10: 81-111. doi: 10.3934/jmd.2016.10.81

Effective decay of multiple correlations in semidirect product actions

Ioannis Konstantoulas ^1,

1.	Department of Mathematics, University of Utah, 155 S 1400 E, Salt Lake City, UT 84112-0090, United States

Received February 2014 Revised February 2016 Published April 2016

Abstract
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We prove effective decay of certain multiple correlation coefficients for measure preserving, mixing Weyl chamber actions of semidirect products of semisimple groups with $G$-vector spaces. These estimates provide decay for actions in split semisimple groups of higher rank.

Keywords: multiple mixing., Exponential mixing, semidirect products.

Mathematics Subject Classification: Primary: 37A25, 22E5.

Citation: Ioannis Konstantoulas. Effective decay of multiple correlations in semidirect product actions. Journal of Modern Dynamics, 2016, 10: 81-111. doi: 10.3934/jmd.2016.10.81

References:

[1]	M. B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511542749.
[2]	M. B. Bekka and M. Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces, London Mathematical Society Lecture Note Series, 269, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511758898.
[3]	M. Björklund, M. Einsiedler and A. Gorodnik, Effective multiple mixing for semisimple groups, in preparation.
[4]	A. Borel, Linear Algebraic Groups, Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0941-6.
[5]	M. Cowling, U. Haagerup and R. Howe, Almost $L^2$ matrix coefficients, J. Reiner Angew. Math., 387 (1988), 97-110.
[6]	D. Dolgopyat, Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc., 356 (2004), 1637-1689. doi: 10.1090/S0002-9947-03-03335-X.
[7]	M. Einsiedler, G. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math., 177 (2009), 137-212. doi: 10.1007/s00222-009-0177-7.
[8]	R. E. Howe and C. C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal., 32 (1979), 72-96. doi: 10.1016/0022-1236(79)90078-8.
[9]	R. Howe and E. C. Tan, Non Abelian Harmonic Analysis, Universitext, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4613-9200-2.
[10]	J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975.
[11]	A. Katok and R. Spatzier, First cohomology of Anosov actions of higher rank Abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ. Math., 79 (1994), 131-156.
[12]	A. W. Knapp, Representation Theory of Semisimple Groups. An Overview Based on Examples, Princeton Landmarks in Mathematics, 36, Princeton University Press, Princeton, NJ, 1986.
[13]	S. Lang, Real Analysis, Second edition, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983.
[14]	F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A561-A563.
[15]	G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991.
[16]	S. Mozes, Mixing of all orders of Lie groups actions, Inventiones Mathematicae, 107 (1992), 235-241. doi: 10.1007/BF01231889.
[17]	H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Mathematical Journal, 113 (2002), 133-192. doi: 10.1215/S0012-7094-02-11314-3.
[18]	V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Translated from the 1991 Russian original by Rachel Rowen, Pure and Applied Mathematics, 139, Academic Press, Inc., Boston, MA, 1994.
[19]	K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995.
[20]	A. N. Starkov, Dynamical Systems on Homogeneous Spaces, Translated from the 1999 Russian original by the author, Translations of Mathematical Monographs, 190, American Mathematical Society, Providence, RI, 2000.
[21]	M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1975.
[22]	T.-H. D. Hui, Mixing and Certain Integral Point Problems on Semisimple Lie Groups, Ph.D Thesis, Yale University, 1998.
[23]	Z. J. Wang, Uniform pointwise bounds for matrix coefficients of unitary representations on semidirect products, J. Funct. Anal., 267 (2014), 15-79. doi: 10.1016/j.jfa.2014.03.014.
[24]	G. Warner, Harmonic Analysis on Semisimple Lie Groups I, Springer-Verlag, Berlin, 1972.

show all references

References:

[1]	M. B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T), Cambridge University Press, Cambridge, 2008. doi: 10.1017/CBO9780511542749.
[2]	M. B. Bekka and M. Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces, London Mathematical Society Lecture Note Series, 269, Cambridge University Press, Cambridge, 2000. doi: 10.1017/CBO9780511758898.
[3]	M. Björklund, M. Einsiedler and A. Gorodnik, Effective multiple mixing for semisimple groups, in preparation.
[4]	A. Borel, Linear Algebraic Groups, Second edition, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0941-6.
[5]	M. Cowling, U. Haagerup and R. Howe, Almost $L^2$ matrix coefficients, J. Reiner Angew. Math., 387 (1988), 97-110.
[6]	D. Dolgopyat, Limit theorems for partially hyperbolic systems, Trans. Amer. Math. Soc., 356 (2004), 1637-1689. doi: 10.1090/S0002-9947-03-03335-X.
[7]	M. Einsiedler, G. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces, Invent. Math., 177 (2009), 137-212. doi: 10.1007/s00222-009-0177-7.
[8]	R. E. Howe and C. C. Moore, Asymptotic properties of unitary representations, J. Funct. Anal., 32 (1979), 72-96. doi: 10.1016/0022-1236(79)90078-8.
[9]	R. Howe and E. C. Tan, Non Abelian Harmonic Analysis, Universitext, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4613-9200-2.
[10]	J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975.
[11]	A. Katok and R. Spatzier, First cohomology of Anosov actions of higher rank Abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ. Math., 79 (1994), 131-156.
[12]	A. W. Knapp, Representation Theory of Semisimple Groups. An Overview Based on Examples, Princeton Landmarks in Mathematics, 36, Princeton University Press, Princeton, NJ, 1986.
[13]	S. Lang, Real Analysis, Second edition, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983.
[14]	F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A561-A563.
[15]	G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991.
[16]	S. Mozes, Mixing of all orders of Lie groups actions, Inventiones Mathematicae, 107 (1992), 235-241. doi: 10.1007/BF01231889.
[17]	H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Mathematical Journal, 113 (2002), 133-192. doi: 10.1215/S0012-7094-02-11314-3.
[18]	V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Translated from the 1991 Russian original by Rachel Rowen, Pure and Applied Mathematics, 139, Academic Press, Inc., Boston, MA, 1994.
[19]	K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995.
[20]	A. N. Starkov, Dynamical Systems on Homogeneous Spaces, Translated from the 1999 Russian original by the author, Translations of Mathematical Monographs, 190, American Mathematical Society, Providence, RI, 2000.
[21]	M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1975.
[22]	T.-H. D. Hui, Mixing and Certain Integral Point Problems on Semisimple Lie Groups, Ph.D Thesis, Yale University, 1998.
[23]	Z. J. Wang, Uniform pointwise bounds for matrix coefficients of unitary representations on semidirect products, J. Funct. Anal., 267 (2014), 15-79. doi: 10.1016/j.jfa.2014.03.014.
[24]	G. Warner, Harmonic Analysis on Semisimple Lie Groups I, Springer-Verlag, Berlin, 1972.

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