What would you like to understand?

The theorem introduced is Sharkovskii's theorem, specifically the case where $n=3$. It is described as a theorem about the existence of periodic points for a given function, proved using fixed point theory.

Conditions: The theorem is invoked as an existence result about periodic points.; Only the case name $n=3$ is stated in this clip.

The theorem introduced is Sharkovskii's theorem, specifically the case where $n=3$. It is described as a theorem about the existence of periodic points for a given function, proved using fixed point theory.

Conditions: The theorem is invoked as an existence result about periodic points.; Only the case name $n=3$ is stated in this clip.

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 $f$.