Smooth Cocycles over Homogeneous Dynamical Systems.

[摘要] In this thesis, we study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center.First, we consider actions arising from the flows of two commuting elements of the Lie algebra---one nilpotent, and the other semisimple.Second, we consider actions from two commuting unipotent flows that come from an embedded copy of $overline{SL(2,RR)}^{l_{1}} times overline{SL(2,RR)}^{l_{2}}$.In both cases we show that any smooth $RR$-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function.(This is commonly referred to as smooth cocycle rigidity.)These build on results of D. Mieczkowski, where the same is shown for actions on $(SL(2,RR) times SL(2,RR))/G$.These results yield a number of corollaries, particularly for group actions that restrict to either of the two types of aforementioned actions.For example, we prove smooth cocycle rigidity for the action of the upper-triangular (and strictly upper-triangular) subgroup of $SL(n,RR)$ on $SL(n,RR)/G$, for $n geq 3$ (respectively, $n > 3$).We also prove smooth cocycle rigidity for higher-rank abelian actions that restrict to one of the two above mentioned actions.As an application, we use this last result to establish rigidity for smooth time-changes of these higher-rank abelian actions.

[发布机构] University of Michigan

[效力级别] Homogeneous Dynamics

[关键词] Smooth Cocycle Rigidity;Homogeneous Dynamics;First Smooth Cohomology;Representation Theory;Dynamical Systems;Ergodic Theory;Mathematics;Science;Mathematics