American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217
[2]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[3]

Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044
[4]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[5]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
[6]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
[7]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[8]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020108
[9]

Shengbing Deng, Tingxi Hu, Chun-Lei Tang. $ N- $Laplacian problems with critical double exponential nonlinearities. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 987-1003. doi: 10.3934/dcds.2020306
[10]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105

2019 Impact Factor: 0.465

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences