American Institute of Mathematical Sciences

July  2019, 39(7): 3969-4000. doi: 10.3934/dcds.2019160

Cohomological equation and cocycle rigidity of discrete parabolic actions

 a. The MITRE Corporation, McLean, VA 22102, USA b. Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Approved for Public Release; Distribution Unlimited. Case Number 18-1082. The first author's affiliation with The MITRE Corporation is provided for identification purposes only, and is not intended to convey or imply MITRE's concurrence with, or support for, the positions, opinions or viewpoints expressed by the authors. ©2018 The MITRE Corporation. All rights reserved.
1 Based on research supported by NSF grant DMS-1700837

Received  June 2018 Revised  January 2019 Published  April 2019

We study the cohomological equation for discrete horocycle maps on ${\rm SL}(2, \mathbb{R})$ and ${\rm SL}(2, \mathbb{R})\times {\rm SL}(2, \mathbb{R})$ via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of ${\rm SL}(2, \mathbb{R})$. Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of $\operatorname{\mathfrak s\mathfrak l}(2, \mathbb{R})$, and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for ${\rm SL}(2, \mathbb{R})$ in which all cases of irreducible, unitary representations of ${\rm SL}(2, \mathbb{R})$ can be studied simultaneously.

Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.

Citation: James Tanis, Zhenqi Jenny Wang. Cohomological equation and cocycle rigidity of discrete parabolic actions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3969-4000. doi: 10.3934/dcds.2019160
References:
 [1] H. Cartan and R. Takahashi, Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, volume 115. Hermann Paris, 1961. [2] D. Damianovic and A. Katok, Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic actions, Discrete Contin. Dynam. Syst, 13 (2005), 985-1005. doi: 10.3934/dcds.2005.13.985. [3] D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions Ⅰ. KAM method and k actions on the torus, Annals of Mathematics, 172 (2010), 1805-1858. doi: 10.4007/annals.2010.172.1805. [4] D. Damjanovic and A. Katok, Local rigidity of homogeneous parabolic actions: Ⅰ. a model case, Journal of Modern Dynamics, 5 (2011), 203-235. doi: 10.3934/jmd.2011.5.203. [5] D. Damjanovic and J. Tanis, Cocycle rigidity and splitting for some discrete parabolic actions, Discrete and Continuous Dynamical Systems, 34 (2014), 5211-5227. doi: 10.3934/dcds.2014.34.5211. [6] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Mathematical Journal, 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8. [7] L. Flaminio and G. Forni, Equidistribution of nilflows and applications to theta sums, Ergodic Theory and Dynamical Systems, 26 (2006), 409-433. doi: 10.1017/S014338570500060X. [8] L. Flaminio and G. Forni, On effective equidistribution for higher step nilflows, arXiv preprint, arXiv: 1407.3640, 2014. [9] L. Flaminio, G. Forni and J. Tanis, Effective equidistribution of twisted horocycle flows and horocycle maps, Geometric and Functional Analysis, 26 (2016), 1359-1448. doi: 10.1007/s00039-016-0385-4. [10] A. Katok and R. J. Spatzier, Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions, Math. Res. Lett., 1 (1994), 193-202. doi: 10.4310/MRL.1994.v1.n2.a7. [11] A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Mathematical Research Letters, 3 (1996), 191-210. doi: 10.4310/MRL.1996.v3.n2.a6. [12] A. Katok and R. J. Spatzier, First cohomology of anosov actions of higher rank abelian groups and applications to rigidity, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 79 (1994), 131–156. [13] A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math., 216 (1997), 287-314. [14] G. A. Margulis, Finitely-additive invariant measures on Euclidean spaces, Ergodic Theory and Dynamical Systems, 2 (1982), 383-396. doi: 10.1017/S014338570000167X. [15] F. I. Mautner, Unitary representations of locally compact groups II, Annals of Mathematics, 52 (1950), 528-556. doi: 10.2307/1969431. [16] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, J. Mod. Dyn, 1 (2007), 61-92. doi: 10.3934/jmd.2007.1.61. [17] 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. [18] 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. [19] J. Tanis and Z. J. Wang, Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups, Geom. Funct. Anal., 25 (2015), 1956-2020. doi: 10.1007/s00039-015-0351-6. [20] A. F. M. Ter Elst and D. W. Robinson, Elliptic operators on Lie groups, Acta Applicandae Mathematicae, 44 (1996), 133-150. doi: 10.1007/BF00116519. [21] Z. J. Wang, Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups, Geometric and Functional Analysis, 25 (2015), 1956-2020. doi: 10.1007/s00039-015-0351-6. [22] Z. J. Wang, Cocycle Rigidity of Partially Hyperbolic Actions, Cocycle rigidity of partially hyperbolic actions, 2017. [23] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9488-4.

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References:
 [1] H. Cartan and R. Takahashi, Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes, volume 115. Hermann Paris, 1961. [2] D. Damianovic and A. Katok, Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic actions, Discrete Contin. Dynam. Syst, 13 (2005), 985-1005. doi: 10.3934/dcds.2005.13.985. [3] D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions Ⅰ. KAM method and k actions on the torus, Annals of Mathematics, 172 (2010), 1805-1858. doi: 10.4007/annals.2010.172.1805. [4] D. Damjanovic and A. Katok, Local rigidity of homogeneous parabolic actions: Ⅰ. a model case, Journal of Modern Dynamics, 5 (2011), 203-235. doi: 10.3934/jmd.2011.5.203. [5] D. Damjanovic and J. Tanis, Cocycle rigidity and splitting for some discrete parabolic actions, Discrete and Continuous Dynamical Systems, 34 (2014), 5211-5227. doi: 10.3934/dcds.2014.34.5211. [6] L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Mathematical Journal, 119 (2003), 465-526. doi: 10.1215/S0012-7094-03-11932-8. [7] L. Flaminio and G. Forni, Equidistribution of nilflows and applications to theta sums, Ergodic Theory and Dynamical Systems, 26 (2006), 409-433. doi: 10.1017/S014338570500060X. [8] L. Flaminio and G. Forni, On effective equidistribution for higher step nilflows, arXiv preprint, arXiv: 1407.3640, 2014. [9] L. Flaminio, G. Forni and J. Tanis, Effective equidistribution of twisted horocycle flows and horocycle maps, Geometric and Functional Analysis, 26 (2016), 1359-1448. doi: 10.1007/s00039-016-0385-4. [10] A. Katok and R. J. Spatzier, Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions, Math. Res. Lett., 1 (1994), 193-202. doi: 10.4310/MRL.1994.v1.n2.a7. [11] A. Katok and A. Kononenko, Cocycles' stability for partially hyperbolic systems, Mathematical Research Letters, 3 (1996), 191-210. doi: 10.4310/MRL.1996.v3.n2.a6. [12] A. Katok and R. J. Spatzier, First cohomology of anosov actions of higher rank abelian groups and applications to rigidity, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 79 (1994), 131–156. [13] A. Katok and R. J. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math., 216 (1997), 287-314. [14] G. A. Margulis, Finitely-additive invariant measures on Euclidean spaces, Ergodic Theory and Dynamical Systems, 2 (1982), 383-396. doi: 10.1017/S014338570000167X. [15] F. I. Mautner, Unitary representations of locally compact groups II, Annals of Mathematics, 52 (1950), 528-556. doi: 10.2307/1969431. [16] D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, J. Mod. Dyn, 1 (2007), 61-92. doi: 10.3934/jmd.2007.1.61. [17] 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. [18] 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. [19] J. Tanis and Z. J. Wang, Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher rank Lie groups, Geom. Funct. Anal., 25 (2015), 1956-2020. doi: 10.1007/s00039-015-0351-6. [20] A. F. M. Ter Elst and D. W. Robinson, Elliptic operators on Lie groups, Acta Applicandae Mathematicae, 44 (1996), 133-150. doi: 10.1007/BF00116519. [21] Z. J. Wang, Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups, Geometric and Functional Analysis, 25 (2015), 1956-2020. doi: 10.1007/s00039-015-0351-6. [22] Z. J. Wang, Cocycle Rigidity of Partially Hyperbolic Actions, Cocycle rigidity of partially hyperbolic actions, 2017. [23] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9488-4.
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