Map Orbit

The forward map orbit of a point under a map is the set

$O_f(x)={x,f(x),f^2(x),...},$

where denotes -fold function composition and is the identity map (Silverman 2010, pp. 2-3). The map orbit is finite iff is a preperiodic point. A periodic point returns to its initial value, while a preperiodic point may have an initial segment before entering a periodic orbit.

See also

Dynamical System, Map Sink, Periodic Orbit, Periodic Point, Preperiodic Point

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References

Silverman, J. H. The Arithmetic of Dynamical Systems. New York: Springer, 2007. https://doi.org/10.1007/978-0-387-69904-2.Silverman, J. H. "Lecture Notes on Arithmetic Dynamics." Arizona Winter School, 2010. https://swc-math.github.io/aws/2010/2010SilvermanNotes.pdf.

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Cite this as:

Weisstein, Eric W. "Map Orbit." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MapOrbit.html

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