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this is a very trivial question, but I couldn't find what the proper name a function of the form $x_{n+1} = f(x_n)$, where $f: X \rightarrow Y$ for some initial $x_0$ is, and what the the sequence $(x_n)_{n=0}^{\infty}$ which generates is called?

As well what a sequence of functions $(f^{(i)})_{i=0}^{\infty}$ for $f:X \rightarrow Y$, and $f^{(m)}$ is the composure of $f$ with itself $m$ times.

Thanks a bunch for your help

sarah
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  • Your question is very unclear. Could you spend a little time expanding upon the contents of your question, perhaps adding something called a "question mark"? – Robin Goodfellow Oct 22 '14 at 23:22

2 Answers2

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This is called a fixed-point iteration.

Adriano
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  • Thanks a bunch! I was wondering whether that was the case, but thought that that was only the method class of numerical methods. – sarah Oct 22 '14 at 23:34
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$Y$ must be the same as $X$ for $x_{n+1} = f(x_n)$ to make sense.

In this case:

  • the sequence $(x_n)$ is called the orbit of $x_0$ under $f$.

  • $f$ is called a map on $X$.

  • a composition of $f$ with itself several times is called an iterate of $f$.

This terminology is used in discrete dynamical systems.

lhf
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