March 2018, 38(3): 989-1006. doi: 10.3934/dcds.2018042

Exponential multiple mixing for some partially hyperbolic flows on products of $ {\rm{PSL}}(2, \mathbb{R})$

USA

Received  May 2016 Revised  September 2017 Published  December 2017

We prove a sharp estimate up to a logarithmic factor on the rate of equidistribution of coordinate horocycle flows on $ Γ \backslash{\rm{PSL}}(2, \mathbb{R})^d$, where $ d ∈ \mathbb{N}_{≥2}$ and $ Γ \subset {\rm{PSL}}(2, \mathbb{R})^d$ is a cocompact and irreducible lattice. As a consequence, we prove exponential multiple mixing for partially hyperbolic coordinate geodesic flows on these manifolds.

Citation: 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
References:
[1]

M. Björklund, M. Einsiedler and A. Gorodnik, Quantitative Multiple Mixing, to appear in J. Eur. Math. Soc. (JEMS)

[2]

M. Brin and Y. Pessin, Flows of frames on manifolds of negative curvature, Uspehi Mat. Nauk., 28 (1973), 209-210.

[3]

M. Brin and Y. Pessin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.

[4]

T. Browning and Ilya Vinogradov, Effective Ratner theorem for $ {\rm{ASL}}(2, \mathbb{R})$ and gaps in $ \sqrt{n}$ modulo 1, J. London Math. Soc., 94 (2016), 61-84.

[5]

S. G. Dani, Kolmogorov automorphisms on homogeneous spaces, Amer. J. Math., 98 (1976), 119-163. doi: 10.2307/2373618.

[6]

S. G. Dani, Spectrum of an affine transformation, Duke Math. J., 44 (1977), 129-155. doi: 10.1215/S0012-7094-77-04407-6.

[7]

D. Dolgopyat, Limit theorems for partially hyperbolic systems, Transactions of the American Mathematical Society, 356 (2004), 1637-1689. doi: 10.1090/S0002-9947-03-03335-X.

[8]

D. Dolgopyat, On Decay of correlations in Anosov flows, Annals of Math., 147 (1998), 357-390. doi: 10.2307/121012.

[9]

L. Flaminio and G. Forni, Invariant Distributions and Time Averages for Horocycle Flows, Duke J. of Math., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.

[10]

L. Flaminio and G. Forni, On effective equidistribution for higher step nilflows, arXiv:1407.3640

[11]

L. FlaminioG. Forni and J. Tanis, Effective equidistribution of twisted horocycle flows and horocycle maps, Geometric and Functional Analysis, 26 (5), 1359-1448.

[12]

A. Gorodnik and R. Spatzier, Exponential mixing of nilmanifold automorphsims, Journal d'Analyse Methematique, 123 (2014), 355-396. doi: 10.1007/s11854-014-0024-7.

[13]

D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of $ {\rm{PSL}}(2, \mathbb R)$, J. Eur. Math. Soc., 11 (2009), 283-313.

[14]

I. Konstantoulas, Effective decay of multiple correlations in semidirect product actions, Journal of Modern Dynamics, 10 (2016), 81-111. doi: 10.3934/jmd.2016.10.81.

[15]

C. Liverani, On Contact Anosov flows, Annals of Math., 159 (2004), 1275-1312. doi: 10.4007/annals.2004.159.1275.

[16]

E. Nelson, Analytic vectors, Annals of Math., 70 (1959), 572-615. doi: 10.2307/1970331.

[17]

A. Strombergsson, An Effective Ratner Equidistribution Result for $ {\rm{SL}}(2,\mathbb R)\ltimes \mathbb R^2$, Duke Math. J., 164 (2015), 843-902. doi: 10.1215/00127094-2885873.

[18]

J. Tanis and P. Vishe, Uniform bounds for period integrals and sparse equidistribution, International Mathematics Research Notices, (2015), 13728-13756. doi: 10.1093/imrn/rnv115.

[19]

J. Tanis, Effective equidistribution for some unipotent flows in $ {\rm{PSL}}(2, \mathbb{R})^k$ mod cocompact irreducible lattice, arXiv:1412.5353v3

[20]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math., 172 (2010), 989-1094. doi: 10.4007/annals.2010.172.989.

[21]

I. Ilya Vinogradov, Effective equidistribution of horocycle lifts, arXiv:1607.04769

show all references

References:
[1]

M. Björklund, M. Einsiedler and A. Gorodnik, Quantitative Multiple Mixing, to appear in J. Eur. Math. Soc. (JEMS)

[2]

M. Brin and Y. Pessin, Flows of frames on manifolds of negative curvature, Uspehi Mat. Nauk., 28 (1973), 209-210.

[3]

M. Brin and Y. Pessin, Partially hyperbolic dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.

[4]

T. Browning and Ilya Vinogradov, Effective Ratner theorem for $ {\rm{ASL}}(2, \mathbb{R})$ and gaps in $ \sqrt{n}$ modulo 1, J. London Math. Soc., 94 (2016), 61-84.

[5]

S. G. Dani, Kolmogorov automorphisms on homogeneous spaces, Amer. J. Math., 98 (1976), 119-163. doi: 10.2307/2373618.

[6]

S. G. Dani, Spectrum of an affine transformation, Duke Math. J., 44 (1977), 129-155. doi: 10.1215/S0012-7094-77-04407-6.

[7]

D. Dolgopyat, Limit theorems for partially hyperbolic systems, Transactions of the American Mathematical Society, 356 (2004), 1637-1689. doi: 10.1090/S0002-9947-03-03335-X.

[8]

D. Dolgopyat, On Decay of correlations in Anosov flows, Annals of Math., 147 (1998), 357-390. doi: 10.2307/121012.

[9]

L. Flaminio and G. Forni, Invariant Distributions and Time Averages for Horocycle Flows, Duke J. of Math., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.

[10]

L. Flaminio and G. Forni, On effective equidistribution for higher step nilflows, arXiv:1407.3640

[11]

L. FlaminioG. Forni and J. Tanis, Effective equidistribution of twisted horocycle flows and horocycle maps, Geometric and Functional Analysis, 26 (5), 1359-1448.

[12]

A. Gorodnik and R. Spatzier, Exponential mixing of nilmanifold automorphsims, Journal d'Analyse Methematique, 123 (2014), 355-396. doi: 10.1007/s11854-014-0024-7.

[13]

D. Kelmer and P. Sarnak, Strong spectral gaps for compact quotients of products of $ {\rm{PSL}}(2, \mathbb R)$, J. Eur. Math. Soc., 11 (2009), 283-313.

[14]

I. Konstantoulas, Effective decay of multiple correlations in semidirect product actions, Journal of Modern Dynamics, 10 (2016), 81-111. doi: 10.3934/jmd.2016.10.81.

[15]

C. Liverani, On Contact Anosov flows, Annals of Math., 159 (2004), 1275-1312. doi: 10.4007/annals.2004.159.1275.

[16]

E. Nelson, Analytic vectors, Annals of Math., 70 (1959), 572-615. doi: 10.2307/1970331.

[17]

A. Strombergsson, An Effective Ratner Equidistribution Result for $ {\rm{SL}}(2,\mathbb R)\ltimes \mathbb R^2$, Duke Math. J., 164 (2015), 843-902. doi: 10.1215/00127094-2885873.

[18]

J. Tanis and P. Vishe, Uniform bounds for period integrals and sparse equidistribution, International Mathematics Research Notices, (2015), 13728-13756. doi: 10.1093/imrn/rnv115.

[19]

J. Tanis, Effective equidistribution for some unipotent flows in $ {\rm{PSL}}(2, \mathbb{R})^k$ mod cocompact irreducible lattice, arXiv:1412.5353v3

[20]

A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math., 172 (2010), 989-1094. doi: 10.4007/annals.2010.172.989.

[21]

I. Ilya Vinogradov, Effective equidistribution of horocycle lifts, arXiv:1607.04769

[1]

Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42.

[2]

Earl Berkson. Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces. Electronic Research Announcements, 2010, 17: 90-103. doi: 10.3934/era.2010.17.90

[3]

Raf Cluckers, Julia Gordon, Immanuel Halupczok. Motivic functions, integrability, and applications to harmonic analysis on $p$-adic groups. Electronic Research Announcements, 2014, 21: 137-152. doi: 10.3934/era.2014.21.137

[4]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757

[5]

Thierry de la Rue. An introduction to joinings in ergodic theory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 121-142. doi: 10.3934/dcds.2006.15.121

[6]

Mirela Domijan, Markus Kirkilionis. Graph theory and qualitative analysis of reaction networks. Networks & Heterogeneous Media, 2008, 3 (2) : 295-322. doi: 10.3934/nhm.2008.3.295

[7]

Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271

[8]

Rafael Tiedra De Aldecoa. Spectral analysis of time changes of horocycle flows. Journal of Modern Dynamics, 2012, 6 (2) : 275-285. doi: 10.3934/jmd.2012.6.275

[9]

Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953

[10]

John R. Tucker. Attractors and kernels: Linking nonlinear PDE semigroups to harmonic analysis state-space decomposition. Conference Publications, 2001, 2001 (Special) : 366-370. doi: 10.3934/proc.2001.2001.366

[11]

C E Yarman, B Yazıcı. A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group. Inverse Problems & Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457

[12]

Luis Barreira, Christian Wolf. Dimension and ergodic decompositions for hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 201-212. doi: 10.3934/dcds.2007.17.201

[13]

Jerrold E. Marsden, Alexey Tret'yakov. Factor analysis of nonlinear mappings: p-regularity theory. Communications on Pure & Applied Analysis, 2003, 2 (4) : 425-445. doi: 10.3934/cpaa.2003.2.425

[14]

Cristian A. Coclici, Jörg Heiermann, Gh. Moroşanu, W. L. Wendland. Asymptotic analysis of a two--dimensional coupled problem for compressible viscous flows. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 137-163. doi: 10.3934/dcds.2004.10.137

[15]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Stability analysis of non-linear plates coupled with Darcy flows. Evolution Equations & Control Theory, 2013, 2 (2) : 193-232. doi: 10.3934/eect.2013.2.193

[16]

Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551

[17]

Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817

[18]

Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure & Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347

[19]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22

[20]

Cristina Lizana, Vilton Pinheiro, Paulo Varandas. Contribution to the ergodic theory of robustly transitive maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 353-365. doi: 10.3934/dcds.2015.35.353

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (17)
  • HTML views (49)
  • Cited by (0)

Other articles
by authors

[Back to Top]