# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020164

## On globally hypoelliptic abelian actions and their existence on homogeneous spaces

 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.

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

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##### 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
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