# 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, Cambridge, 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, 269, Cambridge University Press, Cambridge, 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, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 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), 97-110.  Google Scholar [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.  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), 137-212. 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), 72-96. doi: 10.1016/0022-1236(79)90078-8.  Google Scholar [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.  Google Scholar [10] J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 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), 131-156.  Google Scholar [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.  Google Scholar [13] S. Lang, Real Analysis, Second edition, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 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), A561-A563.  Google Scholar [15] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991.  Google Scholar [16] S. Mozes, Mixing of all orders of Lie groups actions, Inventiones Mathematicae, 107 (1992), 235-241. 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), 133-192. 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, Pure and Applied Mathematics, 139, Academic Press, Inc., Boston, MA, 1994.  Google Scholar [19] K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995.  Google Scholar [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.  Google Scholar [21] M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1975.  Google Scholar [22] T.-H. D. Hui, Mixing and Certain Integral Point Problems on Semisimple Lie Groups, Ph.D Thesis, Yale University, 1998.  Google Scholar [23] Z. J. Wang, Uniform pointwise bounds for matrix coefficients of unitary representations on semidirect products, J. Funct. Anal., 267 (2014), 15-79. doi: 10.1016/j.jfa.2014.03.014.  Google Scholar [24] G. Warner, Harmonic Analysis on Semisimple Lie Groups I, Springer-Verlag, Berlin, 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, Cambridge, 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, 269, Cambridge University Press, Cambridge, 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, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 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), 97-110.  Google Scholar [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.  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), 137-212. 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), 72-96. doi: 10.1016/0022-1236(79)90078-8.  Google Scholar [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.  Google Scholar [10] J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 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), 131-156.  Google Scholar [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.  Google Scholar [13] S. Lang, Real Analysis, Second edition, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 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), A561-A563.  Google Scholar [15] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 17, Springer-Verlag, Berlin, 1991.  Google Scholar [16] S. Mozes, Mixing of all orders of Lie groups actions, Inventiones Mathematicae, 107 (1992), 235-241. 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), 133-192. 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, Pure and Applied Mathematics, 139, Academic Press, Inc., Boston, MA, 1994.  Google Scholar [19] K. Schmidt, Dynamical Systems of Algebraic Origin, Progress in Mathematics, 128, Birkhäuser Verlag, Basel, 1995.  Google Scholar [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.  Google Scholar [21] M. H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1975.  Google Scholar [22] T.-H. D. Hui, Mixing and Certain Integral Point Problems on Semisimple Lie Groups, Ph.D Thesis, Yale University, 1998.  Google Scholar [23] Z. J. Wang, Uniform pointwise bounds for matrix coefficients of unitary representations on semidirect products, J. Funct. Anal., 267 (2014), 15-79. doi: 10.1016/j.jfa.2014.03.014.  Google Scholar [24] G. Warner, Harmonic Analysis on Semisimple Lie Groups I, Springer-Verlag, Berlin, 1972. Google Scholar
 [1] James Tanis. Exponential multiple mixing for some partially hyperbolic flows on products of ${\rm{PSL}}(2, \mathbb{R})$. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 989-1006. doi: 10.3934/dcds.2018042 [2] Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657 [3] Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012 [4] Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315 [5] Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations & Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645 [6] Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217 [7] Xuhui Peng, Jianhua Huang, Yan Zheng. Exponential mixing for the fractional Magneto-Hydrodynamic equations with degenerate stochastic forcing. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4479-4506. doi: 10.3934/cpaa.2020204 [8] Asaf Katz. On mixing and sparse ergodic theorems. Journal of Modern Dynamics, 2021, 17: 1-32. doi: 10.3934/jmd.2021001 [9] Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082 [10] A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195 [11] Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete & Continuous Dynamical Systems, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259 [12] Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817 [13] Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63. [14] Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181 [15] Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079 [16] Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547 [17] Dmitri Scheglov. Absence of mixing for smooth flows on genus two surfaces. Journal of Modern Dynamics, 2009, 3 (1) : 13-34. doi: 10.3934/jmd.2009.3.13 [18] Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175 [19] Jian Li. Localization of mixing property via Furstenberg families. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 725-740. doi: 10.3934/dcds.2015.35.725 [20] Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33

2020 Impact Factor: 0.848