Definition:Periodic Point/Period

This page is about period of periodic point. For other uses, see period.

Definition

Let $T: S \to S$ be an iterated mapping on an arbitrary set $S$.

Let $x \in S$ be a periodic point for $T$ in $S$.

Let $n \in \Z_{>0}$ be the smallest such that $\map {T^n} x = x$.

Then $n$ is the period of $x$ in $T$.

Let $T: \C \to \C$ be the complex function defined such that:

$\forall z \in \C: \map T z = z^3$

Consider the element $i \in \C$.

We have that:

$\ds i^3$

$=$

$\ds -i$

$\ds \paren {-i}^3$

$=$

$\ds i$

where $i$ is the imaginary unit:

$i^2 = -1$

Hence $i$ is a periodic point of period $2$ whose periodic orbit is $\set {i, -i}$.