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periodic point (Definition)

Let $f:X\to X$ be a function and $f^n$ its $n$ th iteration. A point $x$ is called a periodic point of period $n$ of $f$ if it is a fixed point of $f^n$ The least $n$ for which $x$ is a fixed point of $f^n$ is called prime period or least period.

If $f$ is a function mapping $\mathbb{R}$ to $\mathbb{R}$ or $\mathbb{C}$ to $\mathbb{C}$ then a periodic point $x$ of prime period $n$ is called hyperbolic if $|(f^n)'(x)|\neq 1$ attractive if $|(f^n)'(x)|<1$ and repelling if $|(f^n)'(x)|>1$




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Also defines:  hyperbolic periodic point, attractive periodic point, repelling periodic point, least period, prime period
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Cross-references: fixed point, point, iteration, function
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This is version 11 of periodic point, born on 2002-06-04, modified 2002-12-19.
Object id is 3026, canonical name is PeriodicPoint.
Accessed 9215 times total.

Classification:
AMS MSC26A18 (Real functions :: Functions of one variable :: Iteration)

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