• Previous Article
    The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations
  • DCDS Home
  • This Issue
  • Next Article
    Weighted topological and measure-theoretic entropy
July 2019, 39(7): 3923-3940. doi: 10.3934/dcds.2019158

The twisted cohomological equation over the geodesic flow

Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Received  April 2018 Revised  January 2019 Published  April 2019

Fund Project: Based on research supported by NSF grant DMS-1700837

We study the twisted cohomoligical equation over the geodesic flow on $ SL(2, \mathbb{R} )/\Gamma $. We characterize the obstructions to solving the twisted cohomological equation, construct smooth solution and obtain the tame Sobolev estimates for the solution, i.e, there is finite loss of regularity (with respect to Sobolev norms) between the twisted coboundary and the solution. We also give a tame splittings for non-homogeneous cohomological equations. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for partially hyperbolic actions in products of rank-one groups in future works.

Citation: Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158
References:
[1]

D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions. I. KAM method and $ \mathbb{Z} ^k $ actions on the torus, Annals of Mathematics, 172 (2010), 1805-1858. doi: 10.4007/annals.2010.172.1805.

[2]

D. Damjanovic and A. Katok, Local rigidity of homogeneous parabolic actions: I. A model case, J. Modern Dyn., 5 (2011), 203-235. doi: 10.3934/jmd.2011.5.203.

[3]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.

[4]

R. Howe and C. C. Moore, Asymptotic properties of unitary representations, J. Func. Anal., 32 (1979), Kluwer Acad., 72–96. doi: 10.1016/0022-1236(79)90078-8.

[5]

G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, 1991. doi: 10.1007/978-3-642-51445-6.

[6]

F. I. Mautner, Unitary representations of locally compact groups, II, Ann. of Math., (2) 52 (1950), 528–556. doi: 10.2307/1969431.

[7]

D. Mieczkowski, The Cohomological Equation and Representation Theory, Ph.D thesis, The Pennsylvania State University, 2006.

[8]

F. A. Ramirez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups, Journal of Modern Dynamics, 3 (2009), 335-357. doi: 10.3934/jmd.2009.3.335.

[9]

D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Mathematical Monographs, 1991.

[10]

J. Tanis, The cohomological equation and invariant distributions for horocycle maps, Ergodic Theory and Dynamical systems, 34 (2014), 299-340. doi: 10.1017/etds.2012.125.

[11]

Z. J. Wang, Various smooth rigidity examples in$SL(2, \mathbb{R})\times\cdots SL(2, \mathbb{R})/\Gamma $, in preparation.

[12]

Z. J. Wang, The twisted cohomological equation over the partially hyperbolic flow, submitted, arXiv: 1809.04672

[13]

R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984. doi: 10.1007/978-1-4684-9488-4.

show all references

References:
[1]

D. Damjanovic and A. Katok, Local Rigidity of Partially Hyperbolic Actions. I. KAM method and $ \mathbb{Z} ^k $ actions on the torus, Annals of Mathematics, 172 (2010), 1805-1858. doi: 10.4007/annals.2010.172.1805.

[2]

D. Damjanovic and A. Katok, Local rigidity of homogeneous parabolic actions: I. A model case, J. Modern Dyn., 5 (2011), 203-235. doi: 10.3934/jmd.2011.5.203.

[3]

L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math J., 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8.

[4]

R. Howe and C. C. Moore, Asymptotic properties of unitary representations, J. Func. Anal., 32 (1979), Kluwer Acad., 72–96. doi: 10.1016/0022-1236(79)90078-8.

[5]

G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, 1991. doi: 10.1007/978-3-642-51445-6.

[6]

F. I. Mautner, Unitary representations of locally compact groups, II, Ann. of Math., (2) 52 (1950), 528–556. doi: 10.2307/1969431.

[7]

D. Mieczkowski, The Cohomological Equation and Representation Theory, Ph.D thesis, The Pennsylvania State University, 2006.

[8]

F. A. Ramirez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups, Journal of Modern Dynamics, 3 (2009), 335-357. doi: 10.3934/jmd.2009.3.335.

[9]

D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Mathematical Monographs, 1991.

[10]

J. Tanis, The cohomological equation and invariant distributions for horocycle maps, Ergodic Theory and Dynamical systems, 34 (2014), 299-340. doi: 10.1017/etds.2012.125.

[11]

Z. J. Wang, Various smooth rigidity examples in$SL(2, \mathbb{R})\times\cdots SL(2, \mathbb{R})/\Gamma $, in preparation.

[12]

Z. J. Wang, The twisted cohomological equation over the partially hyperbolic flow, submitted, arXiv: 1809.04672

[13]

R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984. doi: 10.1007/978-1-4684-9488-4.

[1]

Bassam Fayad, Raphaël Krikorian. Rigidity results for quasiperiodic SL(2, R)-cocycles. Journal of Modern Dynamics, 2009, 3 (4) : 479-510. doi: 10.3934/jmd.2009.3.479

[2]

Artur Avila, Thomas Roblin. Uniform exponential growth for some SL(2, R) matrix products. Journal of Modern Dynamics, 2009, 3 (4) : 549-554. doi: 10.3934/jmd.2009.3.549

[3]

Artur Avila. Density of positive Lyapunov exponents for quasiperiodic SL(2, R)-cocycles in arbitrary dimension. Journal of Modern Dynamics, 2009, 3 (4) : 631-636. doi: 10.3934/jmd.2009.3.631

[4]

Russell Johnson, Mahesh G. Nerurkar. On $SL(2, R)$ valued cocycles of Hölder class with zero exponent over Kronecker flows. Communications on Pure & Applied Analysis, 2011, 10 (3) : 873-884. doi: 10.3934/cpaa.2011.10.873

[5]

Ser Peow Tan, Yan Loi Wong and Ying Zhang. The SL(2, C) character variety of a one-holed torus. Electronic Research Announcements, 2005, 11: 103-110.

[6]

Julie Déserti. Jonquières maps and $SL(2;\mathbb{C})$-cocycles. Journal of Modern Dynamics, 2016, 10: 23-32. doi: 10.3934/jmd.2016.10.23

[7]

Samuel C. Edwards. On the rate of equidistribution of expanding horospheres in finite-volume quotients of SL(2, ${\mathbb{C}}$). Journal of Modern Dynamics, 2017, 11: 155-188. doi: 10.3934/jmd.2017008

[8]

Anne-Sophie de Suzzoni. Consequences of the choice of a particular basis of $L^2(S^3)$ for the cubic wave equation on the sphere and the Euclidean space. Communications on Pure & Applied Analysis, 2014, 13 (3) : 991-1015. doi: 10.3934/cpaa.2014.13.991

[9]

Dmitry Tamarkin. Quantization of Poisson structures on R^2. Electronic Research Announcements, 1997, 3: 119-120.

[10]

Michael Usher. Floer homology in disk bundles and symplectically twisted geodesic flows. Journal of Modern Dynamics, 2009, 3 (1) : 61-101. doi: 10.3934/jmd.2009.3.61

[11]

C.P. Walkden. Solutions to the twisted cocycle equation over hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 935-946. doi: 10.3934/dcds.2000.6.935

[12]

Guji Tian, Qi Wang, Chao-Jiang Xu. $C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1023-1039. doi: 10.3934/dcds.2016.36.1023

[13]

Imed Bachar, Habib Mâagli. Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 171-188. doi: 10.3934/dcdss.2019012

[14]

Myeongju Chae, Soonsik Kwon. Global well-posedness for the $L^2$-critical Hartree equation on $\mathbb{R}^n$, $n\ge 3$. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1725-1743. doi: 10.3934/cpaa.2009.8.1725

[15]

Giorgio Fusco. Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1807-1841. doi: 10.3934/cpaa.2017088

[16]

Giorgio Fusco, Francesco Leonetti, Cristina Pignotti. On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 725-742. doi: 10.3934/dcds.2017030

[17]

Michał Kowalczyk, Yong Liu, Frank Pacard. Towards classification of multiple-end solutions to the Allen-Cahn equation in $\mathbb{R}^2$. Networks & Heterogeneous Media, 2012, 7 (4) : 837-855. doi: 10.3934/nhm.2012.7.837

[18]

J. Colliander, M. Keel, Gigliola Staffilani, H. Takaoka, T. Tao. Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 665-686. doi: 10.3934/dcds.2008.21.665

[19]

Nicola Sansonetto, Daniele Sepe. Twisted isotropic realisations of twisted Poisson structures. Journal of Geometric Mechanics, 2013, 5 (2) : 233-256. doi: 10.3934/jgm.2013.5.233

[20]

A. Kononenko. Twisted cocycles and rigidity problems. Electronic Research Announcements, 1995, 1: 26-34.

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (26)
  • HTML views (88)
  • Cited by (0)

Other articles
by authors

[Back to Top]