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A point has least or prime period $m$ if it returns to itself after $m$ iterations and does not return earlier at any smaller positive iterate. This excludes trivial cases where a point already repeats with a smaller period.

Conditions: $m$ is a positive integer.; $k$ ranges over integers with $1\le k<m$.; $p$ is a periodic point of $f$.

A point has least or prime period $m$ if it returns to itself after $m$ iterations and does not return earlier at any smaller positive iterate. This excludes trivial cases where a point already repeats with a smaller period.

Conditions: $m$ is a positive integer.; $k$ ranges over integers with $1\le k<m$.; $p$ is a periodic point of $f$.

Saying a point $p$ is fixed by $f^m$ means that applying the function $f$ exactly $m$ times to $p$ returns $p$ itself. This rephrases the definition of a periodic point of period $m$ as a fixed point of the $m$-fold iterate of $f$.

Conditions: $f$ is a function.; $m$ is a positive integer.; $p$ is a point in the domain of $f$.

Saying a point $p$ is fixed by $f^m$ means that applying the function $f$ exactly $m$ times to $p$ returns $p$ itself. This rephrases the definition of a periodic point of period $m$ as a fixed point of the $m$-fold iterate of $f$.

Conditions: $f$ is a function.; $m$ is a positive integer.; $p$ is a point in the domain of $f$.