I was hoping to figure this one out on my own. There's this nice paper by Avila on various "subtractive" Euclidean algorithms. Here is one he attributes to Viggo Brun:
$$ (x,y,z) \mapsto \text{sort}(x-y, y, z) $$
and I even read Brun's original article where he shows a periodic orbit of length 19 for the three numbers:
$$ (\sqrt[3]{4}, \sqrt[3]{2}, 1) \mapsto \big(\sqrt[3]{2} \;,\; 1 \;,\ \sqrt[3]{4}- \sqrt[3]{2} \big) \mapsto \dots $$
This one repeats after 19 steps, but actually not. I checked on a computer (and checked my French) after 19 steps the three digits are proportional to that in the first step. So there is periodicity that way.
When is this thing periodic? If it is periodic, there is $3 \times 3$ matrix encoding that sequence and the eigenvalues must be a roots of the characteristic polynomial, which is a cubic.
Conversely, if I get three numbers which are roots to some polynomial $$ x^3 + ax + b = 0 $$
and if I run the Brun continued fraction with the three roots, $(r_1, r_2, r_3)$ so I always get a periodic sequence? This makes sense if all three roots $r_1, r_2, r_3 \in \mathbb{R}$ are real numbers.
Alternatively we could try to adjoin $\sqrt[3]{n}$ and get the real cubic extension $K = \mathbb{Q}(\sqrt[3]{n})$ and then $1, \sqrt[3]{2}, \sqrt[3]{4}$ span the ring of integer, $\mathcal{O}_K$. We an ask if this sequence is periodic.
$$ (n^{2/3}, n^{1/3}, 1) \mapsto \big(n^{2/3}- n^{1/3}, n^{1/3}, 1\big ) \mapsto \dots $$
Even in the quadratic case, I don't think I've read carefully the argument that periodic continued fractions must be quadratic irrationalities $a + b\sqrt{D}$ with $a,b \in \mathbb{Q}$. Perhaps I could read the relevant part of Khinchin or other resource.
A somewhat precise question
Let $(x,y,z) \in \mathbb{R}P^2$ be three real numbers (up to proportion) and Brun continued fraction algorithm:
$$ T: (x,y,z) \mapsto \text{sort}(x-y, y, z) $$
what are the closed orbits of this map? I have no idea how to make this precise. The traditional continued fraction algorithm can also be made subtrative:
$$ T_0: (x,y) \mapsto \text{sort}(x-y, y) $$
and we can record how many times the coordinates get switched in the process - $(12)$ if $x > y$ and $(21)$ if $y > x$ - and this leads to a continued fraction of sorts. The statement is then:
Every periodic continued fraction represents a quadratic irrational number and every quadratic irrational number can be represented by a periodic continued fraction.
Khinchin Continued Fractions Thm 28 (on p48)
