How to determine the orbits of points under the tripling map $f(x)=3x\bmod 1$?

Question

Let $f$ be the tripling map $f(x) = 3x \mod(1)$. Determine the complete orbit of the points $\frac{1}{8}$ and $\frac{1}{72}$. Indicate whether each of these points is periodic, eventually periodic, or neither.

I've asked my professor for plenty of help on this matter, but she can't explain this too me at all and makes the whole matter even more confusing. Is there any help out there to better explain this type of problem that can lead me to the correct answer?

Answer 1

I suppose $f$ is a map $\Bbb{R} \rightarrow \Bbb{R}$. First you should consider what happens to irrational resp. rational numbers. The$\mod(1)$ expression creates an equivalence relation $x\sim y \Leftrightarrow x-y \in \Bbb {Z}$. Can an irrational number become rational? (i.e can an irrational belong to the same equivalence class as a rational?). Now consider a rational number of the form $\frac{p}{q}<1$ and $\gcd(p,q)=1$. The numbers in the orbit of this number are all of the form $\frac{r}{q}<1$. Show that sooner or later a value will hit a previous value (use the pigeonhole principle). What happens after a previous value has been "hit"?

Answer 2

Although not necessary for this problem, I recommend that you familiarize yourself with some computational tool to help you explore these types of problems. In Mathematica or even WolframAlpha, you could just enter the following:

NestList[Mod[3 #, 1] &, 1/72, 10]

It's now crystal clear that $$\frac{1}{72} \rightarrow \frac{1}{24} \rightarrow \frac{1}{8} \rightarrow \frac{3}{8} \rightarrow \frac{1}{8}$$ and that you've now reached a loop.

Of course, there are many such tools available today and it's easier to answer questions here if we know that you are using some specific software. I would hope your class encourages something along these lines.