Effective decay of multiple correlations in semidirect product actions

Previous Article
Invariant distributions for homogeneous flows and affine transformations
JMD Home
This Volume
Next Article
Horocycle flows for laminations by hyperbolic Riemann surfaces and Hedlund's theorem

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

Abstract
Related Papers

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.
[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).

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.
[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).

[1]	James Tanis. Exponential multiple mixing for some partially hyperbolic flows on products of $ {\rm{PSL}}(2, \mathbb{R})$. Discrete & Continuous Dynamical Systems - A, 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 - A, 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]	Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082
[7]	A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195
[8]	Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259
[9]	Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817
[10]	Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63.
[11]	Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181
[12]	Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079
[13]	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
[14]	Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175
[15]	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
[16]	Jian Li. Localization of mixing property via Furstenberg families. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 725-740. doi: 10.3934/dcds.2015.35.725
[17]	Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33
[18]	Makoto Mori. Higher order mixing property of piecewise linear transformations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 915-934. doi: 10.3934/dcds.2000.6.915
[19]	Benoît Saussol. Recurrence rate in rapidly mixing dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 259-267. doi: 10.3934/dcds.2006.15.259
[20]	François Blanchard, Wen Huang. Entropy sets, weakly mixing sets and entropy capacity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 275-311. doi: 10.3934/dcds.2008.20.275

2018 Impact Factor: 0.295

American Institute of Mathematical Sciences