Definition:Periodic Point

Definition

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

Let $x \in S$ be a point in $S$ such that:

$\exists n \in \Z_{>0}: \map {T^n} x = x$

where $\map {T^n} x$ is defined iteratively as:

$\map {T^n} x := \begin {cases} x & : n = 0 \\ \map T {\map {T^{n - 1} } x} & : n > 0 \end {cases}$

Then $x$ is a periodic point for $T$.

Period

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 $T$.

Periodic Orbit

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

The periodic orbit of $x$ is the set:

$\set {x, \map T x, \map {T^2} x, \ldots, \map {T^n} x}$

Examples

Complex Cube Function

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 $\set {i, -i}$.

Also see

• Results about periodic points can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): periodic point
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): periodic point