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.
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Asaf Karagila
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Shay
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5Sometimes "orbit", particularly if the function is invertible. – Ethan Bolker Oct 29 '18 at 22:01
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1But careful, the word orbit means a lot of things and must be used properly, depending on the context. – Riccardo Ceccon Oct 29 '18 at 22:13
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@Ethan: How can it be orbit if it doesn't include $f^{-1}(a)$ and its iterates? – Asaf Karagila Oct 29 '18 at 22:39
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2"Closure" is a commonly used, and very appropriate, term for this set. It is the smallest set closed under $f$ such that $a$ is in that set. – Asaf Karagila Oct 29 '18 at 22:40
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@AsafKaragila: Yours is the clearest answer to me. Would you mind posting it as an answer so I can select it? – Shay Oct 30 '18 at 14:28
2 Answers
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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.
gt6989b
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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 $f$.
Asaf Karagila
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