Suppose we have $f(z) = (z-r_1)\cdots(z-r_n)$, $|r_j| = 1$. According to the Lucas-Gauss theorem, all of the zeros of $f'$ lie in the convex hull of the $r_j$, but I discovered some behavior I don't understand while playing with the applet at http://demonstrations.wolfram.com/SendovsConjecture/.
Take, for example, the case $n = 5$, and arrange the $r_j$ in "general position" on the unit circle (so that roots of $f'$ do not coincide), taking $r_1 = 1 = e^0$. Now let $r_1$ vary continuously from $e^0$ to $e^{2\pi}$ by increasing its argument. You will find various things happening to the roots $s_1,s_2,s_3,s_4$ of $f'$; of course, when $r_1$ returns to its original position, the roots will coincide with their starting positions. However, it may be that $s_1 \mapsto s_2$, $s_2 \mapsto s_3$, $s_3 \mapsto s_4$, and $s_4 \mapsto s_1$. I have also seen $s_1 \mapsto s_1$, and $s_2 \mapsto s_3$, $s_3 \mapsto s_4$, $s_4 \mapsto s_2$.
Apparently, the act of rotating one of the $r_j$ counterclockwise one revolution is a nontrivial group action on the $s_j$. I don't understand this, or really know what to search for. What's happening here? How can I tell a priori what will happen to the $s_j$ under such an action, given the $r_j$? If this is in the literature, where can I find it?
Edit:
Here's a horrible depiction of an example of order 2:
The blue dots are zeros of $f$; the orange dots are zeros of $f'$. The big red circle shows the motion of $r_1 = 1$, and the other four red curves show how the orange dots behave.
Fix $4$ of the original roots, and let the $5$th move over all of $\Bbb C$. Then $P_\alpha = (X - \alpha)Q(X)$ where $Q$ is a fixed polynomial of degree $4$.
Most of the time, $P_\alpha'$ has $4$ distinct roots. Let $\{\beta_i\}$ be the roots of the discriminant $\Delta(\alpha)$ of $P_\alpha'$ (a polynomial in $\alpha$ of degree $6$, almost always). Then when $\alpha = \beta_i$, $P'\alpha$ has multiple roots. If $\alpha$ makes one small loop around one of those values, you will see two (or more, if $\beta_i$ has multiplicity) very close roots of $P'_\alpha$ being switched.
If you pick a special reference point $\alpha_0 \notin \{\beta_i\}$, and fix an ordering on the roots of $P_{\alpha_0}'$, then you can pick for each $\beta_i$ a loop from $\alpha_0$ to itself that loops only around that $\beta_i$ and describe its monodromy as a particular transposition (or a bigger cycle, in case $\beta_i$ has higher multiplicity) on the roots of $P_{\alpha_0}'$. Since those loops freely generate $\pi_1(\Bbb C - \{\beta_i\})$, you can express the original unit loop as a combination of those (up to conjugacy) and compute the corresponding permutation.
If you want a more general picture, you will need to move more roots, and ultimately, you have to describe $\pi_1(\Bbb C^5 - V(\Delta))$ (or $\pi_1((\Bbb C^5 - V(\Delta)) \cap \Bbb U^5)$ if you want to stay on the unit circle) where $\Delta$ is the discriminant of the derivative of the generic degree $5$ polynomial in term of its roots. And also, everything is prettier if you replace $\Bbb C$ with the Riemann sphere so you can discuss things as one of the root goes to infinity (this lessens the degree of $P$ by $1$).