On globally hypoelliptic abelian actions and their existence on homogeneous spaces

Danijela Damjanovic, ,Royal Institute of Technology, Stockholm, Sweden, James Tanis, Zhenqi Jenny Wang, ,The MITRE Corporation, McLean, VA 22102, USA, ,Michigan State University, East Lansing, MI 48824, USA
2019 Discrete and Continuous Dynamical Systems. Series A  
We define globally hypoelliptic smooth 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 R k actions (k ≥ 2) on
more » ... omogeneous spaces globally hypoelliptic actions exist only on nilmanifolds. We obtain a partial result towards this conjecture: we show non-existence of globally hypoelliptic R 2 actions on homogeneous spaces G/Γ, with at least one quasi-unipotent generator, where G = SL(n, R). We also show that the same type of actions on solvmanifolds are smoothly conjugate to homogeneous actions on nilmanifolds. 2010 Mathematics Subject Classification. Primary: 37C15, 37C85, 37D20.
doi:10.3934/dcds.2020164 fatcat:tg72yn67vzfkrjntrzy6e73wke