Map Cycle
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)
(1)
Y_2 = G(Y_1)
(2)
Y_(n-1) = G(Y_(n-2))
(3)
Y_0 = G(Y_(n-1)).
(4)
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.
See also
Fixed Point, MapExplore with Wolfram|Alpha
WolframAlpha
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Cite this as:
Weisstein, Eric W. "Map Cycle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MapCycle.html