American Institute of Mathematical Sciences

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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

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, (2008).  doi: 10.1017/CBO9780511542749.  Google Scholar

[2]

M. B. Bekka and M. Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces,, London Mathematical Society Lecture Note Series, (2000).  doi: 10.1017/CBO9780511758898.  Google Scholar

[3]

M. Björklund, M. Einsiedler and A. Gorodnik, Effective multiple mixing for semisimple groups,, in preparation., ().   Google Scholar

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A. Borel, Linear Algebraic Groups,, Second edition, (1991).  doi: 10.1007/978-1-4612-0941-6.  Google Scholar

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M. Cowling, U. Haagerup and R. Howe, Almost $L^2$ matrix coefficients,, J. Reiner Angew. Math., 387 (1988), 97.   Google Scholar

[6]

D. Dolgopyat, Limit theorems for partially hyperbolic systems,, Trans. Amer. Math. Soc., 356 (2004), 1637.  doi: 10.1090/S0002-9947-03-03335-X.  Google Scholar

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M. Einsiedler, G. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces,, Invent. Math., 177 (2009), 137.  doi: 10.1007/s00222-009-0177-7.  Google Scholar

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R. E. Howe and C. C. Moore, Asymptotic properties of unitary representations,, J. Funct. Anal., 32 (1979), 72.  doi: 10.1016/0022-1236(79)90078-8.  Google Scholar

[9]

R. Howe and E. C. Tan, Non Abelian Harmonic Analysis,, Universitext, (1992).  doi: 10.1007/978-1-4613-9200-2.  Google Scholar

[10]

J. E. Humphreys, Linear Algebraic Groups,, Graduate Texts in Mathematics, (1975).   Google Scholar

[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.   Google Scholar

[12]

A. W. Knapp, Representation Theory of Semisimple Groups. An Overview Based on Examples,, Princeton Landmarks in Mathematics, (1986).   Google Scholar

[13]

S. Lang, Real Analysis,, Second edition, (1983).   Google Scholar

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F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant,, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978).   Google Scholar

[15]

G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), (1991).   Google Scholar

[16]

S. Mozes, Mixing of all orders of Lie groups actions,, Inventiones Mathematicae, 107 (1992), 235.  doi: 10.1007/BF01231889.  Google Scholar

[17]

H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants,, Duke Mathematical Journal, 113 (2002), 133.  doi: 10.1215/S0012-7094-02-11314-3.  Google Scholar

[18]

V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory,, Translated from the 1991 Russian original by Rachel Rowen, (1991).   Google Scholar

[19]

K. Schmidt, Dynamical Systems of Algebraic Origin,, Progress in Mathematics, (1995).   Google Scholar

[20]

A. N. Starkov, Dynamical Systems on Homogeneous Spaces,, Translated from the 1999 Russian original by the author, (1999).   Google Scholar

[21]

M. H. Taibleson, Fourier Analysis on Local Fields,, Princeton University Press, (1975).   Google Scholar

[22]

T.-H. D. Hui, Mixing and Certain Integral Point Problems on Semisimple Lie Groups,, Ph.D Thesis, (1998).   Google Scholar

[23]

Z. J. Wang, Uniform pointwise bounds for matrix coefficients of unitary representations on semidirect products,, J. Funct. Anal., 267 (2014), 15.  doi: 10.1016/j.jfa.2014.03.014.  Google Scholar

[24]

G. Warner, Harmonic Analysis on Semisimple Lie Groups I,, Springer-Verlag, (1972).   Google Scholar

show all references

References:
[1]

M. B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T),, Cambridge University Press, (2008).  doi: 10.1017/CBO9780511542749.  Google Scholar

[2]

M. B. Bekka and M. Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces,, London Mathematical Society Lecture Note Series, (2000).  doi: 10.1017/CBO9780511758898.  Google Scholar

[3]

M. Björklund, M. Einsiedler and A. Gorodnik, Effective multiple mixing for semisimple groups,, in preparation., ().   Google Scholar

[4]

A. Borel, Linear Algebraic Groups,, Second edition, (1991).  doi: 10.1007/978-1-4612-0941-6.  Google Scholar

[5]

M. Cowling, U. Haagerup and R. Howe, Almost $L^2$ matrix coefficients,, J. Reiner Angew. Math., 387 (1988), 97.   Google Scholar

[6]

D. Dolgopyat, Limit theorems for partially hyperbolic systems,, Trans. Amer. Math. Soc., 356 (2004), 1637.  doi: 10.1090/S0002-9947-03-03335-X.  Google Scholar

[7]

M. Einsiedler, G. Margulis and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces,, Invent. Math., 177 (2009), 137.  doi: 10.1007/s00222-009-0177-7.  Google Scholar

[8]

R. E. Howe and C. C. Moore, Asymptotic properties of unitary representations,, J. Funct. Anal., 32 (1979), 72.  doi: 10.1016/0022-1236(79)90078-8.  Google Scholar

[9]

R. Howe and E. C. Tan, Non Abelian Harmonic Analysis,, Universitext, (1992).  doi: 10.1007/978-1-4613-9200-2.  Google Scholar

[10]

J. E. Humphreys, Linear Algebraic Groups,, Graduate Texts in Mathematics, (1975).   Google Scholar

[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.   Google Scholar

[12]

A. W. Knapp, Representation Theory of Semisimple Groups. An Overview Based on Examples,, Princeton Landmarks in Mathematics, (1986).   Google Scholar

[13]

S. Lang, Real Analysis,, Second edition, (1983).   Google Scholar

[14]

F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant,, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978).   Google Scholar

[15]

G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), (1991).   Google Scholar

[16]

S. Mozes, Mixing of all orders of Lie groups actions,, Inventiones Mathematicae, 107 (1992), 235.  doi: 10.1007/BF01231889.  Google Scholar

[17]

H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants,, Duke Mathematical Journal, 113 (2002), 133.  doi: 10.1215/S0012-7094-02-11314-3.  Google Scholar

[18]

V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory,, Translated from the 1991 Russian original by Rachel Rowen, (1991).   Google Scholar

[19]

K. Schmidt, Dynamical Systems of Algebraic Origin,, Progress in Mathematics, (1995).   Google Scholar

[20]

A. N. Starkov, Dynamical Systems on Homogeneous Spaces,, Translated from the 1999 Russian original by the author, (1999).   Google Scholar

[21]

M. H. Taibleson, Fourier Analysis on Local Fields,, Princeton University Press, (1975).   Google Scholar

[22]

T.-H. D. Hui, Mixing and Certain Integral Point Problems on Semisimple Lie Groups,, Ph.D Thesis, (1998).   Google Scholar

[23]

Z. J. Wang, Uniform pointwise bounds for matrix coefficients of unitary representations on semidirect products,, J. Funct. Anal., 267 (2014), 15.  doi: 10.1016/j.jfa.2014.03.014.  Google Scholar

[24]

G. Warner, Harmonic Analysis on Semisimple Lie Groups I,, Springer-Verlag, (1972).   Google Scholar

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