On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Danijela Damjanovic; James Tanis; Zhenqi Jenny Wang

doi:10.3934/dcds.2020164

Previous Article
Rigidity of random group actions
DCDS Home
This Issue
Next Article
Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions

doi: 10.3934/dcds.2020164

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Danijela Damjanovic ^1,, , James Tanis ^2,†, and Zhenqi Jenny Wang ^3,

1.	Royal Institute of Technology, Stockholm, Sweden
2.	The MITRE Corporation, McLean, VA 22102, USA
3.	Michigan State University, East Lansing, MI 48824, USA

^* Corresponding author: Danijela Damjanovic

† The 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 author. ©2019 The MITRE Corporation. ALL RIGHTS RESERVED

Received June 2019 Revised January 2020 Published March 2020

Fund Project: The first author is supported by Swedish Research Council grant VR-2015-04644. The third author is supported by NSF grant DMS-1346876. Approved for Public Release; Distribution Unlimited. Public Release Case Number 19-2033

We define globally hypoelliptic smooth $ \mathbb R^k $ actions as actions whose leafwise Laplacian along the orbit foliation is a globally hypoelliptic differential operator. When $ k = 1 $, strong global rigidity is conjectured by Greenfield-Wallach and Katok: every globally hypoelliptic flow is smoothly conjugate to a Diophantine flow on the torus. The conjecture has been confirmed for all homogeneous flows on homogeneous spaces [9]. In this paper we conjecture that among homogeneous $ \mathbb R^k $ actions ($ k\ge 2 $) on homogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic $ \mathbb R^2 $ actions on homogeneous spaces $ G/\Gamma $, with at least one quasi-unipotent generator, where $ G = SL(n, \mathbb R) $. We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds.

Keywords: Global hypoellipticity, abelian group actions, homogeneous ergodic actions.

Mathematics Subject Classification: Primary: 37C15, 37C85, 37D20.

Citation: Danijela Damjanovic, James Tanis, Zhenqi Jenny Wang. On globally hypoelliptic abelian actions and their existence on homogeneous spaces. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020164

References:

[1]	L. Auslander, An exposition of the structure of solvmanifolds. Ⅱ. $G$-induced flows, Bull. Amer. Math. Soc, 79 (1973) 262–285. doi: 10.1090/S0002-9904-1973-13139-8. Google Scholar
[2]	A. Avila, B. Fayad and A. Kocsard, On manifolds supporting distributionally uniquely ergodic diffeomorphisms, J. Differential Geometry, 99 (2015) 191–213. doi: 10.4310/jdg/1421415561. Google Scholar
[3]	C. Chen and C. Chi, Hypoelliptic vector fields and almost periodic motion on the Torus $T^n$, Commun. Partial Differential Equations, 25 (2000), 337-354. doi: 10.1080/03605300008821516. Google Scholar
[4]	J. Cygan and L. Richardson, $D$-harmonic distributions and global hypoellipticity on nilmanifolds, Pacific J. Math., 147 (1991), 29-46. doi: 10.2140/pjm.1991.147.29. Google Scholar
[5]	J. Cygan and L. Richardson, Globally hypoelliptic systems of vector fields on nilmanifolds, J. Funct. Anal., 77 (1988), 364-371. doi: 10.1016/0022-1236(88)90093-6. Google Scholar
[6]	D. Damjanović, Actions with globally hypoelliptic Laplacian and rigidity, J. Anal. Math., 129 (2016), 139-163. doi: 10.1007/s11854-016-0018-8. Google Scholar
[7]	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
[8]	L. Flaminio and G. Forni, On the cohomological equation for nilflows, J. Mod. Dyn., 1 (2007), 37-60. doi: 10.3934/jmd.2007.1.37. Google Scholar
[9]	L. Flaminio, G. Forni and F. Rodriguez Hertz, Invariant distributions for homogenenous flows and affine transformations, J. Mod. Dyn., 10 (2016), 33-79. doi: 10.3934/jmd.2016.10.33. Google Scholar
[10]	L. Flaminio and Pa ternian, Linearization of cohomology-free vector fields, Discrete Contin. Dyn. Syst., 29 (2011), 1031-1039. doi: 10.3934/dcds.2011.29.1031. Google Scholar
[11]	G. Forni, On the Greenfield-Wallach and Katok conjectures in dimension three, in Geometric and Probabilistic Structures in Dynamics, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008,197–213. Google Scholar
[12]	S. Greenfield and N. Wallach, Remarks on global hypoellipticity, Trans. Amer. Math. Soc., 183 (1973), 153-164. doi: 10.1090/S0002-9947-1973-0400313-1. Google Scholar
[13]	S. Greenfield and N. Wallach, Globally hypoelliptic vector fields, Topology, 12 (1973), 247-254. doi: 10.1016/0040-9383(73)90011-6. Google Scholar
[14]	A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, in Proc. Sympos. Pure Math., Smooth Ergodic Theory and Its Applications, 69, Amer. Math. Soc., Providence, RI, 2001,107–173. Google Scholar
[15]	A. Katok and R. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ. Math., (1994), 131–156. doi: 10.1007/BF02698888. Google Scholar
[16]	D. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinai's Moscow Seminar on Dynamical Systems, Adv. Math. Sci., 171, Amer. Math. Soc., Providence, RI, 1996,141–172. doi: 10.1090/trans2/171/11. Google Scholar
[17]	D. Kleinbock, N. Shah and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, in Handbook of Dynamical Systems, 1A, North-Holland, Amsterdamm, 2002,813–930. doi: 10.1016/S1874-575X(02)80013-3. Google Scholar
[18]	A. Kocsard, Cohomologically rigid vector fields: The Katok conjecture in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165-1182. doi: 10.1016/j.anihpc.2008.07.005. Google Scholar
[19]	E. Lindenstrauss and B. Weiss, On sets invariant under the action of the diagonal group, Ergodic Theory Dynam. Systems, 21 (2001), 1481-1500. doi: 10.1017/S0143385701001717. Google Scholar
[20]	G. W. Mackey, The Theory of Unitary Group Representations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. Google Scholar
[21]	G. A. Margulis, Discrete subgroups of semisimple Lie groups, Results in Mathematics and Related Areas, 17, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-51445-6. Google Scholar
[22]	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. Google Scholar
[23]	D. Mieczkowski, The Cohomological Equation and Representation Theory, Ph.D thesis, Pennsylvania State University, 2006. Google Scholar
[24]	G. Prasad and M. S. Raghunatan, Cartan subgroups and lattices in semi-simple groups, Ann. of Math. (2), 96 (1972), 296–317. doi: 10.2307/1970790. Google Scholar
[25]	C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compositio Math., 23 (1971), 115-122. Google Scholar
[26]	F. Rodriguez Hertz and J. Rodriguez Hertz, Cohomology free systems and the first Betti number, Discrete Contin. Dyn. Syst., 15 (2006), 193-196. doi: 10.3934/dcds.2006.15.193. Google Scholar
[27]	A. N. Starkov, Dynamical Systems on Homogeneous Spaces, Translations of Mathematical Monographs, 190, American Mathematical Society, Providence, RI, 2000. Google Scholar
[28]	J. Tanis, The cohomological equation and invariant distributions for horocycle maps, Ergodic Theory Dynam. Systems, 34 (2014), 299-340. doi: 10.1017/etds.2012.125. Google Scholar
[29]	J. Tanis and Z. J. Wang, Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups, Geom. Funct. Anal., 25 (2015), 1956-2020. doi: 10.1007/s00039-015-0351-6. Google Scholar
[30]	Z. J. Wang, Cocycle rigidity of abelian partially hyperbolic actions, Israel J. Math., 225 (2018), 147-191. doi: 10.1007/s11856-018-1653-9. Google Scholar
[31]	R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9488-4. Google Scholar

show all references

References:

[1]	L. Auslander, An exposition of the structure of solvmanifolds. Ⅱ. $G$-induced flows, Bull. Amer. Math. Soc, 79 (1973) 262–285. doi: 10.1090/S0002-9904-1973-13139-8. Google Scholar
[2]	A. Avila, B. Fayad and A. Kocsard, On manifolds supporting distributionally uniquely ergodic diffeomorphisms, J. Differential Geometry, 99 (2015) 191–213. doi: 10.4310/jdg/1421415561. Google Scholar
[3]	C. Chen and C. Chi, Hypoelliptic vector fields and almost periodic motion on the Torus $T^n$, Commun. Partial Differential Equations, 25 (2000), 337-354. doi: 10.1080/03605300008821516. Google Scholar
[4]	J. Cygan and L. Richardson, $D$-harmonic distributions and global hypoellipticity on nilmanifolds, Pacific J. Math., 147 (1991), 29-46. doi: 10.2140/pjm.1991.147.29. Google Scholar
[5]	J. Cygan and L. Richardson, Globally hypoelliptic systems of vector fields on nilmanifolds, J. Funct. Anal., 77 (1988), 364-371. doi: 10.1016/0022-1236(88)90093-6. Google Scholar
[6]	D. Damjanović, Actions with globally hypoelliptic Laplacian and rigidity, J. Anal. Math., 129 (2016), 139-163. doi: 10.1007/s11854-016-0018-8. Google Scholar
[7]	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
[8]	L. Flaminio and G. Forni, On the cohomological equation for nilflows, J. Mod. Dyn., 1 (2007), 37-60. doi: 10.3934/jmd.2007.1.37. Google Scholar
[9]	L. Flaminio, G. Forni and F. Rodriguez Hertz, Invariant distributions for homogenenous flows and affine transformations, J. Mod. Dyn., 10 (2016), 33-79. doi: 10.3934/jmd.2016.10.33. Google Scholar
[10]	L. Flaminio and Pa ternian, Linearization of cohomology-free vector fields, Discrete Contin. Dyn. Syst., 29 (2011), 1031-1039. doi: 10.3934/dcds.2011.29.1031. Google Scholar
[11]	G. Forni, On the Greenfield-Wallach and Katok conjectures in dimension three, in Geometric and Probabilistic Structures in Dynamics, Contemp. Math., 469, Amer. Math. Soc., Providence, RI, 2008,197–213. Google Scholar
[12]	S. Greenfield and N. Wallach, Remarks on global hypoellipticity, Trans. Amer. Math. Soc., 183 (1973), 153-164. doi: 10.1090/S0002-9947-1973-0400313-1. Google Scholar
[13]	S. Greenfield and N. Wallach, Globally hypoelliptic vector fields, Topology, 12 (1973), 247-254. doi: 10.1016/0040-9383(73)90011-6. Google Scholar
[14]	A. Katok, Cocycles, cohomology and combinatorial constructions in ergodic theory, in Proc. Sympos. Pure Math., Smooth Ergodic Theory and Its Applications, 69, Amer. Math. Soc., Providence, RI, 2001,107–173. Google Scholar
[15]	A. Katok and R. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ. Math., (1994), 131–156. doi: 10.1007/BF02698888. Google Scholar
[16]	D. Kleinbock and G. A. Margulis, Bounded orbits of nonquasiunipotent flows on homogeneous spaces, in Sinai's Moscow Seminar on Dynamical Systems, Adv. Math. Sci., 171, Amer. Math. Soc., Providence, RI, 1996,141–172. doi: 10.1090/trans2/171/11. Google Scholar
[17]	D. Kleinbock, N. Shah and A. Starkov, Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, in Handbook of Dynamical Systems, 1A, North-Holland, Amsterdamm, 2002,813–930. doi: 10.1016/S1874-575X(02)80013-3. Google Scholar
[18]	A. Kocsard, Cohomologically rigid vector fields: The Katok conjecture in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1165-1182. doi: 10.1016/j.anihpc.2008.07.005. Google Scholar
[19]	E. Lindenstrauss and B. Weiss, On sets invariant under the action of the diagonal group, Ergodic Theory Dynam. Systems, 21 (2001), 1481-1500. doi: 10.1017/S0143385701001717. Google Scholar
[20]	G. W. Mackey, The Theory of Unitary Group Representations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. Google Scholar
[21]	G. A. Margulis, Discrete subgroups of semisimple Lie groups, Results in Mathematics and Related Areas, 17, Springer-Verlag, Berlin, 1991. doi: 10.1007/978-3-642-51445-6. Google Scholar
[22]	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. Google Scholar
[23]	D. Mieczkowski, The Cohomological Equation and Representation Theory, Ph.D thesis, Pennsylvania State University, 2006. Google Scholar
[24]	G. Prasad and M. S. Raghunatan, Cartan subgroups and lattices in semi-simple groups, Ann. of Math. (2), 96 (1972), 296–317. doi: 10.2307/1970790. Google Scholar
[25]	C. Pugh and M. Shub, Ergodic elements of ergodic actions, Compositio Math., 23 (1971), 115-122. Google Scholar
[26]	F. Rodriguez Hertz and J. Rodriguez Hertz, Cohomology free systems and the first Betti number, Discrete Contin. Dyn. Syst., 15 (2006), 193-196. doi: 10.3934/dcds.2006.15.193. Google Scholar
[27]	A. N. Starkov, Dynamical Systems on Homogeneous Spaces, Translations of Mathematical Monographs, 190, American Mathematical Society, Providence, RI, 2000. Google Scholar
[28]	J. Tanis, The cohomological equation and invariant distributions for horocycle maps, Ergodic Theory Dynam. Systems, 34 (2014), 299-340. doi: 10.1017/etds.2012.125. Google Scholar
[29]	J. Tanis and Z. J. Wang, Cohomological equation and cocycle rigidity of parabolic actions in some higher-rank Lie groups, Geom. Funct. Anal., 25 (2015), 1956-2020. doi: 10.1007/s00039-015-0351-6. Google Scholar
[30]	Z. J. Wang, Cocycle rigidity of abelian partially hyperbolic actions, Israel J. Math., 225 (2018), 147-191. doi: 10.1007/s11856-018-1653-9. Google Scholar
[31]	R. J. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Mathematics, 81, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9488-4. Google Scholar

[1]	Federico Rodriguez Hertz. Global rigidity of certain Abelian actions by toral automorphisms. Journal of Modern Dynamics, 2007, 1 (3) : 425-442. doi: 10.3934/jmd.2007.1.425
[2]	John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443
[3]	Woochul Jung, Keonhee Lee, Carlos Morales, Jumi Oh. Rigidity of random group actions. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020130
[4]	Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.
[5]	Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
[6]	A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.
[7]	Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
[8]	Anatole Katok, Federico Rodriguez Hertz. Arithmeticity and topology of smooth actions of higher rank abelian groups. Journal of Modern Dynamics, 2016, 10: 135-172. doi: 10.3934/jmd.2016.10.135
[9]	H. Bercovici, V. Niţică. Cohomology of higher rank abelian Anosov actions for Banach algebra valued cocycles. Conference Publications, 2001, 2001 (Special) : 50-55. doi: 10.3934/proc.2001.2001.50
[10]	Salvatore Cosentino, Livio Flaminio. Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds. Journal of Modern Dynamics, 2015, 9: 305-353. doi: 10.3934/jmd.2015.9.305
[11]	Danijela Damjanovic and Anatole Katok. Local rigidity of actions of higher rank abelian groups and KAM method. Electronic Research Announcements, 2004, 10: 142-154.
[12]	Danijela Damjanovic, Anatole Katok. Local rigidity of homogeneous parabolic actions: I. A model case. Journal of Modern Dynamics, 2011, 5 (2) : 203-235. doi: 10.3934/jmd.2011.5.203
[13]	Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365
[14]	Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[15]	Jean-Paul Thouvenot. The work of Lewis Bowen on the entropy theory of non-amenable group actions. Journal of Modern Dynamics, 2019, 15: 133-141. doi: 10.3934/jmd.2019016
[16]	Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146
[17]	Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487
[18]	David Mieczkowski. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. Journal of Modern Dynamics, 2007, 1 (1) : 61-92. doi: 10.3934/jmd.2007.1.61
[19]	Felipe A. Ramírez. Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups. Journal of Modern Dynamics, 2009, 3 (3) : 335-357. doi: 10.3934/jmd.2009.3.335
[20]	Danijela Damjanović. Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions. Journal of Modern Dynamics, 2007, 1 (4) : 665-688. doi: 10.3934/jmd.2007.1.665

American Institute of Mathematical Sciences

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

On globally hypoelliptic abelian actions and their existence on homogeneous spaces

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors