Divergent trajectories of flows on homogeneous spaces and Diophantine approximation.

*(English)*Zbl 0578.22012Let \(G\) be a connected Lie group and \(\Gamma\) be a lattice in \(G\); that is, \(\Gamma\) is a discrete subgroup of \(G\) such that \(G/\Gamma\) admits a finite \(G\)-invariant measure. Let \(\{g_t\}\) be a one-parameter subgroup of \(G\). The action of \(\{g_t\}\) on \(G/\Gamma\) (on the left) is studied. At present time the behavior of “typical” trajectories is satisfactorily understood. In general, however, it is very difficult to describe the behavior of exceptional trajectories.

The author assumes \(G/\Gamma\) to be non-compact and investigates a special class of such exceptional trajectories: “divergent” trajectories. A trajectory is said to be divergent if eventually it leaves every compact subset of \(G/\Gamma\). There is explained how the divergence of trajectories is related to a question involving diophantine approximation for certain systems of linear forms. In particular, using some results of number theory the author proves the following assertion. Let \(G=\mathrm{SL}(n,\mathbb R)\) and \(\Gamma=\mathrm{SL}(n,\mathbb Z)\), and \(g_t\) is a one-parameter subgroup of the form \(\text{diag}(e^{-t},\ldots, e^{-t},e^{\lambda t},\ldots, e^{\lambda t})\). Then the set of points on bounded trajectories has full Hausdorff dimension (equal to that of \(G/\Gamma)\).

The author assumes \(G/\Gamma\) to be non-compact and investigates a special class of such exceptional trajectories: “divergent” trajectories. A trajectory is said to be divergent if eventually it leaves every compact subset of \(G/\Gamma\). There is explained how the divergence of trajectories is related to a question involving diophantine approximation for certain systems of linear forms. In particular, using some results of number theory the author proves the following assertion. Let \(G=\mathrm{SL}(n,\mathbb R)\) and \(\Gamma=\mathrm{SL}(n,\mathbb Z)\), and \(g_t\) is a one-parameter subgroup of the form \(\text{diag}(e^{-t},\ldots, e^{-t},e^{\lambda t},\ldots, e^{\lambda t})\). Then the set of points on bounded trajectories has full Hausdorff dimension (equal to that of \(G/\Gamma)\).

Reviewer: G. A. Margulis (Moskva)

##### MSC:

22E40 | Discrete subgroups of Lie groups |

43A85 | Harmonic analysis on homogeneous spaces |

37C10 | Dynamics induced by flows and semiflows |

11J99 | Diophantine approximation, transcendental number theory |