Divergent trajectories of flows on homogeneous spaces and Diophantine approximation By S. G. Dani*) at Bombay Let G be a connected Lie group and à be a lattice in G; that is, à is a discrete sub- group of G such that G/à admits a finite G-invariant measure. Let {g t } teP be a one- parameter subgroup of G. The action of {g f } on G/T (on the left) induces a flow on G/Ã. The ergodic theory of these flows is extensively studied and, at least from a certain point of view, satisfactorily understood (cf. [6] and its references). Thus, for instance, it is possible to determine, in terms of the position of {g t } in G relative to Ã, whether the flowadmits dense trajectories {g t ÷ à | t ^ 0}, where ÷ e G, and whether a generic trajectory (either with respect to the measure or topologically) is dense in G/Ã. In general, however, there exist exceptional trajectories which are not dense, but to describe their set is a very difficult task; for an arbitrary one-parameter subgroup this is known only when G is a nilpotent Lie group (cf. [18] for that case and [16] for results on horocycle flows). In this paper we assume G/à to be non-compact and investigate a special class of such exceptional trajectories: 'divergent' trajectories. A trajectory is said to be divergent if eventually it leaves every compact subset of G/à (cf. § l for precise definition). In §§ 2 and 3 we also get some results on bounded trajectories of certain flows. It was proved by G. A. Margulis in [21] that if G = S£(«,/R) and r = SL(n,I) and {g t } is a one-parameter subgroup consisting of unipotent elements then there are no divergent trajectories (cf. [11] and [14] for stronger results). A similar Situation can be seen to hold if all the eigenvalues of g p tefl are of absolute value l (cf. Proposition 2. 6). However if g 1 (or any g p /ÖÏ) has some eigenvalue ë with \ë\ ö l then there exist at least certain Obvious' divergent trajectories. For instance, if G = S£(2, IR), à = 5£(2, Z) and g f = diag(e~ f , e*), then the trajectory starting from any point of ÑÃ/Ã, where P is the subgroup consisting of all upper triangul r matrices in G, is divergent for simple geometric reasons. We call these degenerate divergent trajectories (cf. § 2 for details). In §2 we also consider the one-parameter subgroups of G of the form diagOr',...,*-', **...,**), *) Supported in part by NSF grant MCS-8108814 (A02). Brought to you by | University of Iowa Libraries Authenticated Download Date | 5/30/15 1:14 PM