Let $T:\Bbb Z_2\to\Bbb Z_2$ be a bijective 2-adic isometry satisfying $T(0)=0$ and $T(2x)=2T(x)$ and $T(1)=\overline{01}_2$.
For all $y\in\Bbb N\subset\Bbb Z_2$ such that $y<x$, let $T(y)$ be eventually cyclic with period $2$.
Question
Is it possible for there to be some positive integer $x\in\Bbb N\subset\Bbb Z_2$ such that $T(x)$ never becomes cyclic with period $2$?
Motivation
I'm trying to better understand the rules of isometries in the context of integers embedded in the 2-adic space. An example would be useful.
My poor attempt - such as it is
I've tried to create a counterexample but failed. But I also feel like my attempt wasn't a strong attempt.
I can see how the $\overline{01}_2$ string has the potential to constrain possible choices for $T(x)$, especially given that $T(2^ny):n\in\Bbb N$ are all already picked, and picking $T(x)$ also picks $T(2^mx):m\in\Bbb N$.