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

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?

• you maybe interested in this thread mathoverflow.net/questions/38105/… – Willie Wong Oct 17 '10 at 0:31
• Thanks for the link! Thurston's answer went a bit over my head, but I hadn't realized Sergei Tabachnikov wrote significantly on the topic. I'm taking a seminar class with him at the moment, so I'll probably just try to get him to discuss it. – WWright Oct 17 '10 at 0:41
• Ha, what a coincidence! Talk about being at the right place. :-) – Hans Lundmark Oct 17 '10 at 12:26
• It kind of makes sense now given the subject of all our REU projects. I'll see if I can get him to talk about the topic and if he obliges, I'll tex up some notes and post them. – WWright Oct 17 '10 at 20:59

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.

• Thanks for following up on your earlier post. I fixed a small typo and added a reference to Singer's original article. Maybe it would have been better to append this addendum to your earlier answer. – t.b. Sep 1 '11 at 19:54

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.