1. ## Question on terminology

Let $X$ be a set, and let $f:X \to X$ be any function. I am using $f^n(x)$ as function iteration notation for any $x \in X$ (where $f^0(x) = x$). Do either of these two sets have standard names? Here I am defining them with the names I was giving them, which I would like to swap out for standard ones if they exist:

The set of all elements of $X$ that are in the same "tree" as $x$:
$\text{tree}(x) = \left\{y \in X \mid \exists m,n \in \Bbb{N}, f^n(y) = f^m(x) \right\}$

The set of elements of $X$ that $f$ eventually takes to $x$ (I am calling it a tree rooted at $x$):
$\text{rtree}(x) = \left\{y\in X\mid \exists n \in \Bbb{N}\cup \{0\}, f^n(y) = x \right\}$

2. ## Re: Question on terminology

I am realizing why I was having trouble finding information on this. I have been approaching the topic from a graph theory point of view while typically this type of concept is found when considering dynamical systems. Looking at limit sets, stable and unstable sets, etc., it appears that my "rooted tree at x" is similar to a stable set, only I am not requiring my space to be a topological space, I am not requiring f to be a homomorphism, and I am not terribly concerned with fixed points. I guess my naming convention is as good as any. If anyone has a suggestion on a decent book that might help me grasp the concept as it arises in dynamical systems, I would be grateful. Of course, I will be searching for materials on my own if no one has any ideas for me.

3. ## Re: Question on terminology

Originally Posted by SlipEternal
I am realizing why I was having trouble finding information on this. I have been approaching the topic from a graph theory point of view while typically this type of concept is found when considering dynamical systems. Looking at limit sets, stable and unstable sets, etc., it appears that my "rooted tree at x" is similar to a stable set, only I am not requiring my space to be a topological space, I am not requiring f to be a homomorphism, and I am not terribly concerned with fixed points. I guess my naming convention is as good as any. If anyone has a suggestion on a decent book that might help me grasp the concept as it arises in dynamical systems, I would be grateful. Of course, I will be searching for materials on my own if no one has any ideas for me.
Unless I'm mistaken (which isn't unlikely) your "tree" is just a trajectory and x would be an attractor point.
It's not quite the same as a trajectory of a dynamic system in the symplectic plane but if you plot $f^{(n)}$ vs. $f^{(n-1)}$ I believe it's fairly similar.

This came up in a search and may have some use for what you are doing.

4. ## Re: Question on terminology

A trajectory is something similar to what I am looking for. Using graph theory terminology, given a function $f:X\to X$, define a graph $G(X)$ with $V(G) = X$ and $E(G) = \{xf(x) \mid x \in X\}$. Then $\text{tree}(x)$ is the component containing $x$ and $\text{rtree}(x)$ essentially gives all of the vertices that are "above" $x$ in a tree. This is not quite the same as an attractor point since $x$ is not special. It is any arbitrary point. The more I think about this, the more it seems likely that I am just not using the right terminology when I try to look this up, as it seems a rather natural way to evaluate a function in a graph theory setting.

For points that eventually have limit sets/limit cycles, $\text{tree}(x)$ can be thought of a tree rooted at that cycle. I am more interested in the points that do not have limit sets.

I am also interested in the set $Y = \bigcap_{n=1}^\infty \overline{f^n(X)}$ (here I am using the closure of these sets because after looking at the examples from the topological setting, it appears this may be necessary to ensure that such a limit set exists). I am wondering what $f|_Y$ looks like.