In section 5 of Brun Expansions, Substitutions and Discrete Geometry a ``$d$-dimensional" Brun map is introduced:

$$ T: (a_1, \dots, a_n) \mapsto
(\tfrac{a_1}{a_k}, \dots, \{\tfrac{1}{a_j} \},\dots,\tfrac{a_n}{a_k})$$

where $\{ x\} \equiv x \mod 1$. Maps of this kind usually arise as an accelerated version of the Euclidean algorithm.

We could imagine a map $R: x \mapsto x + a$ with initial value $0$.

Consider the first-return map on the interval $[0,R(0)=a]$. We need to find a number such that

$$ ma < a \mod 1$$

In fact, $m = \lfloor \tfrac{1}{a} \rfloor+1$. If we rescale the interval so that $[0,a]$ is now $[0,1]$. In that case

$$ 0 \mapsto \lfloor \tfrac{1}{a} \rfloor \mod \tfrac{1}{a} \equiv \{ \tfrac{1}{a}\}$$

So our rotation got rescaled to $R_1: x \mapsto x + \{ \tfrac{1}{a}\}$. This renormalization is acting on the set of rotations.

It's likely the Brun expansion arises from a "Euclidean algorithm" on vectors.

Here is another interesting question: Do simple, multi-dimensional generalizations of this continued fraction formula exist?

Google searching has found many interesting variants on the continued fraction algorithm:

The more I read the more ambiguous the term "multidimensional continued fraction" gets.

V. Brun, Algorithmes euclidiens pour trois et quatre nombres, 13th Congr. Math. Scand. Helsinki (1957), 45-64.– Daniel Rust Jul 25 '14 at 16:06