# American Institute of Mathematical Sciences

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

## Effective decay of multiple correlations in semidirect product actions

 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.
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. [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. [3] M. Björklund, M. Einsiedler and A. Gorodnik, Effective multiple mixing for semisimple groups,, in preparation., (). [4] A. Borel, Linear Algebraic Groups,, Second edition, (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. [6] D. Dolgopyat, Limit theorems for partially hyperbolic systems,, Trans. Amer. Math. Soc., 356 (2004), 1637. 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. 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. doi: 10.1016/0022-1236(79)90078-8. [9] R. Howe and E. C. Tan, Non Abelian Harmonic Analysis,, Universitext, (1992). doi: 10.1007/978-1-4613-9200-2. [10] J. E. Humphreys, Linear Algebraic Groups,, Graduate Texts in Mathematics, (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. [12] A. W. Knapp, Representation Theory of Semisimple Groups. An Overview Based on Examples,, Princeton Landmarks in Mathematics, (1986). [13] S. Lang, Real Analysis,, Second edition, (1983). [14] F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant,, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978). [15] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), (1991). [16] S. Mozes, Mixing of all orders of Lie groups actions,, Inventiones Mathematicae, 107 (1992), 235. 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. 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, (1991). [19] K. Schmidt, Dynamical Systems of Algebraic Origin,, Progress in Mathematics, (1995). [20] A. N. Starkov, Dynamical Systems on Homogeneous Spaces,, Translated from the 1999 Russian original by the author, (1999). [21] M. H. Taibleson, Fourier Analysis on Local Fields,, Princeton University Press, (1975). [22] T.-H. D. Hui, Mixing and Certain Integral Point Problems on Semisimple Lie Groups,, Ph.D Thesis, (1998). [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. [24] G. Warner, Harmonic Analysis on Semisimple Lie Groups I,, Springer-Verlag, (1972).

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##### References:
 [1] M. B. Bekka, P. de la Harpe and A. Valette, Kazhdan's Property (T),, Cambridge University Press, (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, (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, (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. [6] D. Dolgopyat, Limit theorems for partially hyperbolic systems,, Trans. Amer. Math. Soc., 356 (2004), 1637. 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. 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. doi: 10.1016/0022-1236(79)90078-8. [9] R. Howe and E. C. Tan, Non Abelian Harmonic Analysis,, Universitext, (1992). doi: 10.1007/978-1-4613-9200-2. [10] J. E. Humphreys, Linear Algebraic Groups,, Graduate Texts in Mathematics, (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. [12] A. W. Knapp, Representation Theory of Semisimple Groups. An Overview Based on Examples,, Princeton Landmarks in Mathematics, (1986). [13] S. Lang, Real Analysis,, Second edition, (1983). [14] F. Ledrappier, Un champ markovien peut être d'entropie nulle et mélangeant,, C. R. Acad. Sci. Paris Sér. A-B, 287 (1978). [15] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups,, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), (1991). [16] S. Mozes, Mixing of all orders of Lie groups actions,, Inventiones Mathematicae, 107 (1992), 235. 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. 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, (1991). [19] K. Schmidt, Dynamical Systems of Algebraic Origin,, Progress in Mathematics, (1995). [20] A. N. Starkov, Dynamical Systems on Homogeneous Spaces,, Translated from the 1999 Russian original by the author, (1999). [21] M. H. Taibleson, Fourier Analysis on Local Fields,, Princeton University Press, (1975). [22] T.-H. D. Hui, Mixing and Certain Integral Point Problems on Semisimple Lie Groups,, Ph.D Thesis, (1998). [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. [24] G. Warner, Harmonic Analysis on Semisimple Lie Groups I,, Springer-Verlag, (1972).
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