Iterating powers of complex integers along axes of symmetry

[Ventrella]

I am exploring the behaviors of complex integers (Gaussian and Eisenstein integers). My understanding is that when a complex integer z with norm >1 is multiplied by itself repeatedly, it creates a series of perfect powers. For instance, the Gaussian integer 1+i generates the series 2i, -2+2i, -4, -4-4i, etc., and the norms are powers of 2. The series generated from such an iteration always lies on a logarithmic spiral, unless the initial value of z is on the real axis, in which case the series follows a line that extends straight along the positive side of the real axis. I hope I stated that correctly :) Here's a nice web page with an interactive tool to explore the behaviors of complex powers:

http://plansoft.org/edu/power.html

The set of Gaussian integers forms 8 axes of symmetry (the real and imaginary axes, and the four diagonals). The set of Eisenstein integers forms 12 axes of symmetry. These axes correspond to the "pie slices" that are isomorphic to each other, given rotations and reflections. If z lies on an axis of symmetry, then I believe the numbers in the series will always lie on an axis of symmetry. Is this assumption correct?

My question concerns the inverse: if z (does not) lie on an axis, will the resulting series of powers (never) lie on an axis? If we do not constrain ourselves to the complex integers, then I believe it is possible that there are certain values of z that will generate series in which a (subset) of numbers lie on an axis - however, in this case, the notion of "axis" has a different meaning. My question pertains to complex integers specifically.

Thank you!

-Jeffrey

 

[fresh_42]

All you have done is considered $z^n=(r\cdot u)^n=r^n\cdot u^n$ with a real number $r>1$ and a complex number $u=e^{i\varphi}$ on the unit circle. So $n \longmapsto r^n$ is of course of exponential growth to the origin, whereas $u^n$ jumps around the circle in steps of $\varphi$ degrees: $n\longmapsto n\cdot\varphi \,.$ Together this is your spirale. Now you can choose $r$ and $\varphi$ in a way, that they will met one of your axis (rational values) or not (irrational values).

 

[Ventrella]

Thank you.

I'm not sure what some of those variables refer to, but I believe you are saying that I can characterize this problem in terms of polar coordinates, which allows me to see that the argument changes in regular steps, and therefore z consistently remains either on or off an axis throughout iteration.
-j

 

[WWGD]

I think there may be an additional exception to the spiral behavior. Points of length 1 will define a circle . The sequence will be infinite iff the initial argument is commensurate to $\pi$. EDIT: Since powers of Complexes are defined in terms of the Complex exponential, you may be able to switch branches to end up in different places. Still, the main branch $Logz$ is the one where Real numbers have argument =0. Obviously, if |z|<1 , the spiral will be inward and will approach $0$.