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Let $A,B$ be some sets with $A\supseteq B$, let $a\in A$ be some element and let $f:A\to B$ be some map. Let the sequence $\left(a_n\right)_{n\in\mathbb{N}_0}$ be defined by $a_0 := a$ and $a_n:= f\left(a_{n-1}\right)$ for all $n\in\mathbb{N}$.

I would like give $\left(a_n\right)_{n\in\mathbb{N}_0}$ a name, something like "the [...] sequence of $f$ with starting value $a$", and I was wondering if such name already exsits in the literature.

Wilfred Montoya
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    "The sequence of iterates of $f$ beginning at $a$" would be widely understood and fairly common. Different writers may vary... – coffeemath Oct 10 '23 at 10:35
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    I've seen orbit used this way. – kimchi lover Oct 10 '23 at 10:39
  • @kimchilover In my opinion orbit conveys the idea of the (unordered) set of values than the sequence of iterates. – coffeemath Oct 10 '23 at 10:45
  • @coffeemath The second paragraph of https://en.wikipedia.org/wiki/Orbit_(dynamics) says something slightly different. – kimchi lover Oct 10 '23 at 10:50
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    @coffeemath: FYI, I've almost always seen "orbit" used in the sense of a sequence, at least in iteration theory. See these papers, for example. Also, maybe this MSE question is relevant. – Dave L. Renfro Oct 10 '23 at 13:32
  • @kimchi lover: FYI, I've almost always seen "orbit" used in the sense of a sequence, at least in iteration theory. See these papers, for example. Also, maybe this MSE question is relevant. [same comment twice because the software limits me to alerting only one username] – Dave L. Renfro Oct 10 '23 at 13:33
  • To all. I'd retract my orbit remarks if it would not leave other comments unexplined. – coffeemath Oct 10 '23 at 16:09
  • @coffeemath: Actually, the MSE question I linked to indicates (in comment(s), answer(s), ...) that an unordered set of values is sometimes meant by "orbit", or at least I thought was the case (I don't remember now). Anyway, dynamical systems stuff is not something I know much about, but some aspects of it overlap with my interests and with mathematicians whose work I'm familiar with (e.g. these google search results) and for those aspects "orbits" are sequences. – Dave L. Renfro Oct 10 '23 at 17:13
  • @DaveL.Renfro I had been thinking of the orbit-stabilizer theorem of group actions, in which unless the acting group has an order it doesn't see like orbits are ordered. Anyway I see now "orbit" is used in several ways. – coffeemath Oct 10 '23 at 19:10

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$\left(a_n\right)_{n\in\mathbb{N}_0}$ ist called the orbit of $a$.

Wilfred Montoya
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