Name for the set of unique $\{a, f(a), f(f(a)), ... \}$?

Question

Is there a name for the unique values produced by recursive function calls? Something like

$f(x) =$ (recursive applications of) $(x \cdot 2) \mod 6$
$f(1) = \{1, 2, 4\}$

Thank you.

Answer 1

One appropriate term would be "The closure of $\{a\}$ under $f$", since $\{f^n(a)\mid n\in\Bbb N\}=\{a,f(a),f(f(a)),\dots\}$ is in fact the smallest set containing $a$ which is closed under $f$.

Of course, $\{a\}$ can be replaced by any subset of the domain of $f$.

Answer 2

These are called fixed point iterates, and the process is called fixed point iteration.

In dynamical systems, the term orbit is used as well, as Ethan Bolker indicated in his comment.