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Non-formally integrable centers admitting an algebraic inverse integrating factor
Exponential multiple mixing for some partially hyperbolic flows on products of $ {\rm{PSL}}(2, \mathbb{R})$
USA |
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.
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. Flaminio, G. 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. Flaminio, G. 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 |
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