Schwarzschild Derivative
Definition1
Let $p$ be a fixed point or a periodic point of the smooth map $f : \mathbb{R} \to \mathbb{R}$.
- $f ' (c) = 0$ being $c$ is called the critical point of $f$.
- If the basin of $p$ includes an interval of infinite length, it is called an infinite basin.
- $\displaystyle S(f)(x) := {{f ''' (x) } \over { f '(x) }} - {{3} \over {2}} \left( {{f ''' (x) } \over { f '(x) }} \right)^2$ is called the Schwartzian derivative of $f$.
- If for all $f ' (x) \ne 0$, $S(f)(x) < 0$ implies that $f$ has a negative Schwartzian.
- $\displaystyle h(x) := {{ax + b} \over {cx + d}}$ is called a Möbius map.
Theorem
- [1]: $h$ is the Möbius map $\iff$ $S(h)(x) = 0$
- [2]: If $f$ and $g$ have a negative Schwartzian, then $f \circ g$ also has a negative Schwartzian.
- [3]: If $f$ has a negative Schwartzian, then $f^{k}$ also has a negative Schwartzian.
- [4]: If $p$ is a fixed point or a periodic point of $f$ with a negative Schwartzian, it means:
- ①: There exists a critical point in the basin of $p$, or
- ②: $p$ has an infinite basin, or
- ③: $p$ is a source.
Yorke. (1996). CHAOS: An Introduction to Dynamical Systems: p132. ↩︎