Orbit (dynamics)

For discrete-time dynamical systems, the orbits are sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces.

Definition

Given a dynamical system (T, M, Φ) with T a group, M a set and Φ the evolution function

${\displaystyle \mathrm{\Phi }:U\to M}$ where ${\displaystyle U\subset T\times M}$ with ${\displaystyle \mathrm{\Phi }(0,x)=x}$

we define

${\displaystyle I(x):=\{t\in T:(t,x)\in U\},}$

then the set

${\displaystyle {\gamma }_x:=\{\mathrm{\Phi }(t,x):t\in I(x)\}\subset M}$

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 ${\displaystyle t\ne 0}$ in ${\displaystyle I(x)}$ such that

${\displaystyle \mathrm{\Phi }(t,x)=x}$.

Real dynamical system

Given a real dynamical system (R, M, Φ), I(x) is an open interval in the real numbers, that is ${\displaystyle I(x)=(t_x^{-},t_x^{+})}$. For any x in M

${\displaystyle {\gamma }_x^{+}:=\{\mathrm{\Phi }(t,x):t\in (0,t_x^{+})\}}$

is called positive semi-orbit through x and

${\displaystyle {\gamma }_x^{-}:=\{\mathrm{\Phi }(t,x):t\in (t_x^{-},0)\}}$

is called negative semi-orbit through x.

Discrete time dynamical system

For a discrete time dynamical system with a time-invariant evolution function ${\displaystyle f}$:

The forward orbit of x is the set :

${\displaystyle {\gamma }_x^{+}\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}\{f^t(x):t\ge 0\}}$

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

${\displaystyle {\gamma }_x^{-}\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}\{f^t(x):t\le 0\}}$

and orbit of x is the set :

${\displaystyle {\gamma }_x\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}{\gamma }_x^{-}\cup {\gamma }_x^{+}}$

where :

• ${\displaystyle f}$ is the evolution function ${\displaystyle f:X\to X}$
• set ${\displaystyle X}$ is the dynamical space,
• ${\displaystyle t}$ is number of iteration, which is natural number and ${\displaystyle t\in T}$
• ${\displaystyle x}$ is initial state of system and ${\displaystyle x\in X}$

General dynamical system

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

In addition, a related term is a bounded orbit, when the set ${\displaystyle G.x}$ is pre-compact inside ${\displaystyle X}$.

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 ${\displaystyle S{L}_3(\mathbb{R})\mathrm{\setminus }S{L}_3(\mathbb{Z})}$ 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.