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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, 1991. doi: 10.1007/978-3-642-51445-6.  Google Scholar [6] F. I. Mautner, Unitary representations of locally compact groups, II, Ann. of Math., (2) 52 (1950), 528–556. doi: 10.2307/1969431.  Google Scholar [7] D. Mieczkowski, The Cohomological Equation and Representation Theory, Ph.D thesis, The Pennsylvania State University, 2006.  Google Scholar [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.  Google Scholar [9] D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Mathematical Monographs, 1991.  Google Scholar [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.  Google Scholar [11] Z. J. Wang, Various smooth rigidity examples in$SL(2, \mathbb{R})\times\cdots SL(2, \mathbb{R})/\Gamma$, in preparation. Google Scholar [12] Z. J. Wang, The twisted cohomological equation over the partially hyperbolic flow, submitted, arXiv: 1809.04672 Google Scholar [13] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984. doi: 10.1007/978-1-4684-9488-4.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [5] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, 1991. doi: 10.1007/978-3-642-51445-6.  Google Scholar [6] F. I. Mautner, Unitary representations of locally compact groups, II, Ann. of Math., (2) 52 (1950), 528–556. doi: 10.2307/1969431.  Google Scholar [7] D. Mieczkowski, The Cohomological Equation and Representation Theory, Ph.D thesis, The Pennsylvania State University, 2006.  Google Scholar [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.  Google Scholar [9] D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Mathematical Monographs, 1991.  Google Scholar [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.  Google Scholar [11] Z. J. Wang, Various smooth rigidity examples in$SL(2, \mathbb{R})\times\cdots SL(2, \mathbb{R})/\Gamma$, in preparation. Google Scholar [12] Z. J. Wang, The twisted cohomological equation over the partially hyperbolic flow, submitted, arXiv: 1809.04672 Google Scholar [13] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1984. doi: 10.1007/978-1-4684-9488-4.  Google Scholar
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