$3x+1$ Conjecture and link with Ergodic Theory I read here https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Lagarias3-23.pdf that the Collatz conjecture is equivalent to $ Q_{\infty}(\mathbb{N}) \subseteq \frac{1}{3} \mathbb{Z} $. Where $Q_{\infty} : \mathbb{Z}_2 \to \mathbb{Z}_2 $ is a continous and measure preserving transformation defined by
$$ x \mapsto Q_{\infty}(x) = \sum_{k=0}^{\infty} \left( T^k(x) \mod 2 \right) 2^k $$
and $T : \mathbb{Z}_2 \to \mathbb{Z}_2 $ is an ergodic map defined by
$$ x \mapsto T(x) =\left\{\begin{matrix}
\frac{x}{2} & \text{if} & x \equiv 0 \mod 2 \\ 
 \frac{3x+1}{2}& \text{if} & x \equiv 1 \mod 2 
\end{matrix}\right. $$
I was wondering how to prove that they are equivalent, and I was wondering if there is any reference I can read to further explore the connection between the Collatz conjecture and the Ergodic Theory.

There's an idea in play here that isn't completely obvious. First, let's talk about the more straightforward part.
Choose a positive integer, say $5$. It has a 2-adic expansion, usually written $...0000101.$, which represents (reading from right-to-left) $1\times2^0 + 0\times2^1 + 1\times2^2 + 0\times2^3 + 0\times 2^4 + \cdots$.
The same number has another binary sequence associated with it, namely, its parity sequence under the (shortcut version of the) Collatz function. Since the number's trajectory is:
$5 \to 8 \to 4 \to 2 \to 1 \to 2  \to 1 \to 2  \to 1 \to 2 \to \cdots$,
we write the parity sequence $1,0,0,0,1,0,1,0,1,0,\ldots$
Now here's the interesting part: We can re-interpret that parity sequence as another 2-adic number! Reversing the digits, to see it the usual way around, it looks like:
$...0101010001.$, or emphasizing the repeating part: $\overline{01}0001.$
This is the 2-adic expansion of the rational number $-\frac{13}3$.
This is the map $Q_\infty$. We have just seen that $Q_\infty(5)=-\frac{13}3$.
Now, the fact that it's a fraction with denominator $3$ reflects the fact that it falls into a pattern repeating "$01$". (Note that $\overline{01}.=-\frac13$) Therefore, the claim that every element of $\mathbb{N}$ has a trajectory eventually reaching $1,2,1,2,\ldots$ is transformed, with this $Q_\infty$ map, into the claim that every element of $\mathbb{N}$ is mapped by $Q_\infty$ to a 2-adic integer with $...0101$ trailing off to the left. Such numbers are precisely the elements of $\frac13\mathbb{Z}\setminus\mathbb{Z}$.
Does this help?
