Notation for orbits in Ergodic Theory

Question

Consider a map $T:X\to X$ from a set to itself. In ergodic theory, we say that the orbit of a point $x\in X$ is the set \begin{equation} \{x,Tx,T^2x,T^3x,\cdots\}.\end{equation} Is there any standard nomenclature for such a set? I expect there is, since for example in group theory there is notation to describe an orbit, but I can't find any ergodic theory analogue.

Answer 1

As far as I am aware, there is no standard notation for orbits in dynamics or ergodic theory. Here are some common notations:

• $\operatorname{orb}(x)$ (as FShrike commented),
• $O(x)$,
• $\mathcal{O}_x(T)$ (this is what I personally use; it signals that $\mathcal{O}(T)$ is the orbit partition w/r/t the transformation $T$, and $\mathcal{O}_x(T)$ is the cell in the orbit partition passing through $x$),
• $\mathbb{Z}x$, similar to how one would denote it in group theory.

In your case the orbit is one sided, to denote this it is common to decorate the above with e.g. a $+$ or $\geq0$.