What is the formal definition of a fixed point of a function?

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.

Fixed-point theorems are important because they have broad applications in topology, chaotic dynamics, and differential equations. Although the definition alone may seem abstract, applying these theorems in simple settings can produce surprisingly strong results.

Conditions: Applies to the mathematical study of fixed-point theory.

A periodic point of period $m$ is a point that returns to itself after $m$ iterations of the function. The video emphasizes that this is recurrence under repeated application of $f$, not the same notion of period used for trigonometric functions.

Conditions: $f$ is a function under consideration.; $m$ is a positive integer counting the number of iterations.; $p$ is a point for which the iterates are defined.

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 $f$.

The lecture introduces least (or prime) period to avoid trivial cases. If a point returns to itself after $m$ iterations, it will also return after any multiple of $m$.

Conditions: $f$ is a function.; $p$ is a periodic point of $f$.