Periodic rotation sequence on a circle

Question

So I was wondering if there is a circle rotation sequence which is periodic. So for example, if a Kangaroo were to jump around a circle in jumps that are of the same length/angle, would the kangaroo ever get back to the same point and start repeating its positions?

Answer 1

To make things much easier, let's take a unit circle $S^1$ (meaning we take a line segment of unit length and glue its two endpoints to obtain a circle). Let $0\leq\theta<1$ be an angle, and $R_\theta:x\mapsto x+\theta (\mod 1)$ be rotation by $\theta$. Then a standard theorem in basic dynamics tells us the following:

1. If $\theta$ is rational, then the orbit $\{x,x+\theta,x+2\theta,...,x+n\theta,...\}$ of any point $x$ is periodic.
2. If $\theta$ is irrational, then the orbit of any point is dense.

Suppose said kangaroo is immortal and has nothing better to do but to circle around a unit circle. Then in the first case it will eventually come back exactly to the point where it started looping around, and in the second case it will jump somewhere arbitrarily close to any point on the circle eventually (of course we need to fix an arbitrarily small patch in the foot of the kangaroo to keep track off etc.).

Answer 2

Your question is very much similar to like the wave model implications of Bohr that clearly mentioned existence of constructive interfering waves in a Hydrogen like species iff the angular momentum is integral multiple of $\frac{h}{2\pi}$.

Similarly. In your question. If kangaroo makes jumps of rational multiples of $\pi$(The angle) , it will suffice to return to the starting point and start all over again.