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# Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function

$S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$

Here is a somewhat more conceptual definition, which justifies the terminology. Define $[f, z, \epsilon]$ to be the cross ratio $[f(z), f(z + \epsilon); f(z + 2\epsilon), f(z + 3\epsilon)]$, and let $[z, \epsilon]$ denote the cross ratio $[z, z + \epsilon, z + 2\epsilon, z + 3\epsilon]$ (in fact this is just 4, but this notation makes my point clearer). One can ask if $[f, z, \epsilon]$ is well approximated by $[z, \epsilon]$; indeed, it turns out that the error is $o(\epsilon)$. So one pursues the second order error term and finds that $[f, z, \epsilon] = [z, \epsilon] - 2 S(f)(z) \epsilon^2 + o(\epsilon^2)$. So the Schwarzian derivative measures the infinitessimal change in cross ratio caused by $f$. In particular, $S(f)$ is identically zero precisely for Mobius transformations.

That's all background. From what I have said so far, the Schwarzian derivative is at best a curiosity. What is not obvious at first glance is the fact that the Schwarzian derivative has magical powers. Here are some examples:

First magical power: The Schwarzian derivative is deeply relevant to one dimensional dynamics, stemming from the fact that it behaves in a specific way under compositions. For example, if $f$ is a smooth function from the unit interval to itself with negative Schwarzian derivative and $n$ critical points, then it has at most n+2 attracting periodic orbits.

Second magical power: It says something profound about the solutions to the Sturm-Liouville equation, $f''(z) + u(z) f(z) = 0$. If $f_1$ and $f_2$ are two linearly independent solutions, then the ratio $g(z) = f_1(z)/f_2(z)$ satisfies $S(g) = 2u$.

Third magical power: The Schwarzian derivative is the unique projectively invariant 1-cocyle for the diffeomorphism group of $\mathbb{R}P^1$. This is probably just a restatement of the conceptual definition I gave above, but I'm not sure; in any event, this gives the Schwarzian derivative a great deal of relevance to conformal field theory (or so I'm told).

I'm sure there are more. I'm wondering if all of these powers can be explained by some underlying geometric principle. They all seem vaguely relevant to each other, but the first power in particular seems very hard to relate to the definition in any obvious way. Does anybody have any insights?