Editing Orbit (dynamics) (section)

== Definition ==
[[File:Simple Harmonic Motion Orbit.gif|right|thumb|300px|Diagram showing the periodic orbit of a mass-spring system in [[simple harmonic motion]]. (Here the velocity and position axes have been reversed from the standard convention in order to align the two diagrams)]]

Given a dynamical system (''T'', ''M'', Φ) with ''T'' a [[group (mathematics)|group]], ''M'' a [[set (mathematics)|set]] and Φ the evolution function

:<math>\Phi: U \to M</math> where <math>U \subset T \times M</math> with <math>\Phi(0,x)=x</math>

we define

:<math>I(x):=\{t \in T : (t,x) \in U \},</math>

then the set

:<math>\gamma_x:=\{\Phi(t,x) : t \in I(x)\} \subset M</math>

is called the '''orbit''' through ''x''. An orbit which consists of a single point is called '''constant orbit'''. A non-constant orbit is called '''closed''' or '''periodic''' if there exists a <math>t\neq 0</math> in <math>I(x)</math> such that 
:<math>\Phi(t, x) = x </math>.

=== Real dynamical system ===

Given a real dynamical system (''R'', ''M'',  Φ), ''I''(''x'') is an open interval in the [[real number]]s, that is <math>I(x) = (t_x^- , t_x^+)</math>. For any ''x'' in ''M''
:<math>\gamma_{x}^{+} := \{\Phi(t,x) : t \in (0,t_x^+)\}</math>
is called '''positive semi-orbit''' through ''x'' and
:<math>\gamma_{x}^{-} := \{\Phi(t,x) : t \in (t_x^-,0)\}</math>
is called '''negative semi-orbit''' through ''x''.

=== Discrete time dynamical system ===
For a discrete time dynamical system with a time-invariant evolution function <math> f </math>:

The '''forward''' orbit of x is the set : 
:<math> \gamma_{x}^{+} \  \overset{\underset{\mathrm{def}}{}}{=}   \     \{ f^{t}(x) : t \ge 0 \} </math>

If the function is invertible, the '''backward''' orbit of x is the set :

:<math>\gamma_{x}^{-} \ \overset{\underset{\mathrm{def}}{}}{=}   \     \{f^{t}(x)  : t \le 0 \} </math>

and '''orbit''' of x is the set :

:<math>\gamma_{x}  \  \overset{\underset{\mathrm{def}}{}}{=}  \  \gamma_{x}^{-} \cup \gamma_{x}^{+} </math>

where :
* <math>f</math> is the evolution function <math>f : X \to X </math>
* set <math>X</math> is the '''dynamical space''',
*<math>t</math> is number of iteration, which is [[natural number]] and <math>t \in T </math>
*<math>x</math> is initial state of system and  <math>x \in X </math>

=== 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.

=== Notes ===

It is often the case that the evolution function can be understood to compose the elements of a [[group (mathematics)|group]], in which case the [[orbit (group theory)|group-theoretic orbits]] of the [[Group action (mathematics)|group action]] are the same thing as the dynamical orbits.