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Cycle (Map)

An $n$-cycle is a finite sequence of points $Y_0$, ..., $Y_{n-1}$ such that, under a Map $G$,

$\displaystyle Y_1$ $\textstyle =$ $\displaystyle G(Y_0)$  
$\displaystyle Y_2$ $\textstyle =$ $\displaystyle G(Y_1)$  
$\displaystyle Y_{n-1}$ $\textstyle =$ $\displaystyle G(Y_{n-2})$  
$\displaystyle Y_0$ $\textstyle =$ $\displaystyle G(Y_{n-1}).$  

In other words, it is a periodic trajectory which comes back to the same point after $n$ iterations of the cycle. Every point $Y_j$ of the cycle satisfies $Y_j = G^n(Y_j)$ and is therefore a Fixed Point of the mapping $G^n$. A fixed point of $G$ is simply a Cycle of period 1.

© 1996-9 Eric W. Weisstein