Editing Orbit (dynamics) (section)

=== General dynamical system ===
For a general dynamical system, especially in homogeneous dynamics, when one has a "nice" group <math>G</math> acting on a probability space <math>X</math> in a measure-preserving way, an orbit <math>G.x \subset X</math> will be called periodic (or equivalently, closed) if the stabilizer <math>Stab_{G}(x)</math> is a lattice inside <math>G</math>.

In addition, a related term is a bounded orbit, when the set <math>G.x</math> is pre-compact inside <math>X</math>.

The classification of orbits can lead to interesting questions with relations to other mathematical areas, for example the Oppenheim conjecture (proved by Margulis) and the Littlewood conjecture (partially proved by Lindenstrauss) are dealing with the question whether every bounded orbit of some natural action on the homogeneous space <math>SL_{3}(\mathbb{R})\backslash SL_{3}(\mathbb{Z})</math> is indeed periodic one, this observation is due to Raghunathan and in different language due to Cassels and Swinnerton-Dyer . Such questions are intimately related to deep measure-classification theorems.