Twisted little plots

As preparation for a project, I wanted to look at the behavior of $x^x$ on the unit circle. For those not familiar with the unit circle, that's on the complex plane, where a complex number is displayed by its "real" part (along the x-axis) and its "imaginary" part (along the y-axis). Numbers with absolute value = 1 appear on a circle in this plane. Each point on the graph represents a single complex number, not two numbers, though it can be broken down into two numbers.

I start with points $x$ on the unit circle (marked in blue), calculate $x^x$, and connect the dots for the results in green. There are a few red lines to guide the eye showing what points on the unit circle map to points on the new curve.

Now since $1 = e^{2 \pi N}$ where $N$ is an integer, there can be a lot of different results. The simplest case is $N=0$, of course. In that case $-1^{-1}=-1$. The point where the curve crosses itself is at $(e^{-\pi/2},0)$.

A bit twisted.

If you're curious what happens when $N=1$ (I was), look at this.

The real curve is smooth; I only used a few points to calculate it which is why it looks jerky. There's a bit of swooping around 0 that doesn't show up at this resolution. To see that, look at the central part. For N=-2, -1, 0, 1, 2, the central part looks like this:

There are things that look like $1$ that don't entirely act like $1$.