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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})$

James Tanis

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.

Keywords: Homogeneous flows, ergodic theory, harmonic analysis analysis.
Mathematics Subject Classification: Primary: 37A17, 37A25; Secondary: 37A45.
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) Google Scholar

[2]

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

[3]

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

[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.   Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[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.   Google Scholar

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[10]

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

[11]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[21]

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

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