I was interesting in seeing how well powers are distributed in a finite field, and I came across a peculiar phenomenon.
I used the prime number $p=1009$, and I did my calculations in the finite field $\mathbb F_{1009}$.
I calculated all the powers $\mu^k$ where $\mu:=123\in \mathbb F_{1009}$, and I draw the following graph (the $x$-axis is $k\in \{1,\ldots,1009\}$ and the $y$-axis is $\mu^k\in \mathbb F_{1009}$):
We can see some "arabesques" on this graph, and my question is: where do they come from ? What is causing those little curves ?
Note that you're seeing the same graph four times because $123^{504}\equiv 1\pmod{1009}$ (so everything repeats itself offset horizontally by half the graph width) and $123^{252}\equiv-1\pmod{1009}$ so the upper half of the graph is just a mirror image of the lower half, slided by a fourth of the graph width. So every pattern comes from the rectangle marked in green here:
The repetition may lead you to perceive random micro-patterns in the graph as more characteristic than they really are because each of them repeats four times.
I'm not sure what you mean by "arabesque", but the most characteristic-looking remaining patterns I can see are the falling curves I've marked with red in the above graph.
These are because $123^{-13}\equiv 2\pmod{1009}$, so whenever you have a dot close to the $x$-axis, its double, quadruple, etc will appear to the left of it separated by horizontal distances of $13$.