Editing Collatz conjecture (section)

===2-adic extension===
The function
<math display="block"> T(x) = \begin{cases} \frac{x}{2} &\text{if } x \equiv 0 \pmod{2}\\ \frac{3x+1}{2} & \text{if } x\equiv 1 \pmod{2} \end{cases}</math>
is well-defined on the ring <math>\mathbb{Z}_2</math> of [[2-adic integers]], where it is continuous and [[Measure-preserving transformation|measure-preserving]] with respect to the 2-adic measure. Moreover, its dynamics is known to be [[Ergodic theory|ergodic]].<ref name="Lagarias (1985)"/>

Define the ''parity vector'' function {{mvar|Q}} acting on <math>\mathbb{Z}_2</math> as
<math display="block"> Q(x) = \sum_{k=0}^{\infty} \left( T^k (x) \mod 2 \right) 2^k .</math>

The function {{mvar|Q}} is a 2-adic [[isometry]].<ref>{{Cite journal|last1=Bernstein|first1=Daniel J.|last2=Lagarias|first2=Jeffrey C.|date=1996|title=The 3''x'' + 1 conjugacy map|journal=[[Canadian Journal of Mathematics]]|language=en|volume=48|issue=6|pages=1154–1169|doi=10.4153/CJM-1996-060-x|doi-access=free|issn=0008-414X}}</ref> Consequently, every infinite parity sequence occurs for exactly one 2-adic integer, so that [[almost all]] trajectories are acyclic in <math>\mathbb{Z}_2</math>.

An equivalent formulation of the Collatz conjecture is that
<math display="block"> Q\left(\mathbb{Z}^{+}\right) \subset \tfrac13 \mathbb{Z}.</math>