📂Dynamics

Schwarzschild Derivative

Definition¹

Let $p$ be a fixed point or a periodic point of the smooth map $f : \mathbb{R} \to \mathbb{R}$.

1. $f ' (c) = 0$ being $c$ is called the critical point of $f$.
2. If the basin of $p$ includes an interval of infinite length, it is called an infinite basin.
3. $\displaystyle S(f)(x) := {{f ''' (x) } \over { f '(x) }} - {{3} \over {2}} \left( {{f ''' (x) } \over { f '(x) }} \right)^2$ is called the Schwartzian derivative of $f$.
4. If for all $f ' (x) \ne 0$, $S(f)(x) < 0$ implies that $f$ has a negative Schwartzian.
5. $\displaystyle h(x) := {{ax + b} \over {cx + d}}$ 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.