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?

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    $\begingroup$ you maybe interested in this thread mathoverflow.net/questions/38105/… $\endgroup$ – Willie Wong Oct 17 '10 at 0:31
  • $\begingroup$ 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. $\endgroup$ – WWright Oct 17 '10 at 0:41
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    $\begingroup$ Ha, what a coincidence! Talk about being at the right place. :-) $\endgroup$ – Hans Lundmark Oct 17 '10 at 12:26
  • $\begingroup$ 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. $\endgroup$ – 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.


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.

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    $\begingroup$ 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. $\endgroup$ – 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.


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