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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 $ \vert(f^n)'(x)\vert\neq 1$, attractive if $ \vert(f^n)'(x)\vert<1$ and repelling if $ \vert(f^n)'(x)\vert>1$.



"periodic point" is owned by mathwizard.
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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
There are 11 references to this entry.

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 7455 times total.

Classification:
AMS MSC: 26A18 (Real functions :: Functions of one variable :: Iteration)
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