Cycle (Map)

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

$$Y_1 = G(Y_0)$$

$$Y_2 = G(Y_1)$$

$$Y_{n-1} = G(Y_{n-2})$$

$$Y_0 = G(Y_{n-1}).$$

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.