**Definition:** Let $r(m)$ denote the residue class $r+m\mathbb{Z}$,
where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$,
let the *class transposition* $\tau_{r_1(m_1),r_2(m_2)}$ be the permutation
of $\mathbb{Z}$ which interchanges $r_1+km_1$ and $r_2+km_2$ for every
$k \in \mathbb{Z}$ and which fixes everything else.

The group $G := \langle \tau_{0(2),1(2)}, \tau_{1(2),4(6)}, \tau_{0(3),4(6)} \rangle$ has 3 orbits on $\mathbb{Z}$ -- one of them consists of the negative integers, and the other two form a partition of the nonnegative integers. The latter partition does look somewhat complicated.

There is computational evidence that the orbits have natural densities, where $0^G$ seems to have density about 0.685, and $2^G$ correspondingly seems to have density about 0.315. Also, both orbits seem to be approximately uniformly distributed on the residue classes (mod $m$), for every positive integer $m$. It seems that differences of consecutive members of an orbit are always odd.

Questions:

Is there an alternative description of the 2 orbits of the group $G$ on $\mathbb{N}_0$ (i.e. not just as "the orbits of $G$ on $\mathbb{N}_0$")?

What are the natural densities of the orbits, if they exist? Are they rational, algebraic or transcendental?

## Background

The Collatz conjecture is equivalent to the assertion that the group $H := \langle \tau_{0(2),1(2)}, \tau_{1(2),2(4)}, \tau_{1(4),2(6)} \rangle$ acts transitively on $\mathbb{N}_0$.

Clearly the action of $G$ on $\mathbb{Z}$ is much easier to understand -- e.g. one can relatively easily count orbits. Nevertheless it is not a trivial case, thus one might hope that its investigation might provide some insights into the action of $H$ as well.

## Data

There are tables of the numbers less than 10000 in any of the 2 orbits of $G$ on $\mathbb{N}_0$: orbit1_10000.txt, orbit2_10000.txt. Larger versions with bound 1000000 rather than 10000 are available as well: orbit1_1000000.txt, orbit2_1000000.txt.