Schwarzian Derivative and One-Dimensional Dynamics - how are they connected?

Question

During the summer, I did an REU where we focused primarily on one-dimensional dynamics and more specifically kneading theory. One thing that I was always confused about is why the Schwarzian derivatives always seem to pop up in discussions of iterated dynamics on the real line. I understand what a Schwarzian derivative is, but I don't see any intuitive reason that it should show up in this area.

I was wondering if anyone could explain or provide me with a reference that makes the appearance of Schwarzian derivatives in one-dimensional dynamics on the real line seem natural.

Another question I have, is there an intuitive motivation for the Schwarzian derivative itself?

Answer 1

I don't have any intuitive reason for your question, but i have a very powerful theorem about the Schwarzian Derivative and Dynamical Systems.

Let $f:[a,b] \to \mathbb{R}$ on $C^3$. Suppose that $f'(x)\neq 0$ for all $x \in [a,b]$. Suppose too that $S(f)(x) <0$ for all $x \in [a.b]$. If $f$ have only a finite number of critical points, then for every $n \in \mathbb{N}$ we have that $f$ have only a finite number of periodic orbits with period $n$.

This is a weak version of theorem, but i look in my books and i give you the strong version in a few days. This theorem that has a relationship between Schwarzian Derivative and Dynamical Systems, i hope that be useful for you.

Answer 2

This is another theorem that has a relationship between Schwarzian Derivative and Dynamical Systems, By Singer.

Let $I$ a close interval and $f:I \to I$ of class $C^3$ with $S(f)(x)<0$ for all $x \in I$, qhere $S(f)(x)$ represent the Schwarzian derivative. If $f$ has $n$ critical points, then $f$ has at most $n+2$ attracting periodic orbits.

This is the full version of theorem. I hope that be useful for you.Regards.

---

Edit:

The Schwarzian derivative was introduced into real dynamics by Singer in

David Singer, Stable Orbits and Bifurcation of Maps of the Interval, SIAM Journal on Applied Mathematics Vol. 35, No. 2 (Sep., 1978), pp. 260–267.