This post makes a focus on a very specific part of that long post. Consider the following map:
$$f: n \mapsto \left\{ \begin{array}{ll} \left \lfloor{n/\sqrt{2}} \right \rfloor & \text{ if } n \text{ even,} \\ \left \lfloor{n\sqrt{2}} \right \rfloor & \text{ if } n \text{ odd.} \end{array} \right.$$

Let $f^{r+1}:=f \circ f^r$, consider the orbit of $n=73$ on the iterations of $f$, i.e. the sequence $f^r(73)$: $$73, 103, 145, 205, 289, 408, 288, 203, 287, 405, 572, 404, 285, 403, 569, 804, 568, 401, \dots$$ Now consider the probability of the $m$th first terms of the sequence $f^r(73)$ to be even: $$p_{0}(m):= \frac{|\{ r<m \ | \ f^r(73) \text{ is even}\}|}{m}.$$ Then $p_1(m):=1-p_0(m)$ is the probability of the $m$th first terms of $f^r(73)$ to be odd.

If we compute the values of $p_i(m)$ for $m=10^{\ell}$, $\ell=1,\dots, 5$, we get something unexpected: $$\scriptsize{ \begin{array}{c|c} \ell & p_0(10^{\ell}) &p_1(10^{\ell}) \newline \hline 1 &0.2&0.8 \newline \hline 2 &0.45&0.55 \newline \hline 3 &0.467&0.533 \newline \hline 4 &0.4700&0.5300 \newline \hline 5 &0.46410&0.53590 \end{array} }$$

Question: Is it true that $f^r(73)$ never reach a cycle and that $p_0(m)$ converges to (say) $\alpha$ close to $46.5\%$. What is its exact value of $\alpha$?

The following picture provides the values of $p_0(m)$ for $100 < m < 20000$:

Note that this phenomenon is not specific to $n=73$, but seems to happen as frequently as $n$ is big, and then, the analogous probability seems to converge to the same $\alpha$. If $n <100$, then it happens for $n=73$ only, but for $n<200$, it happens for $n=73, 103, 104, 105, 107, 141, 145, 146, 147, $ $ 148, 149, 151, 152, 153, 155, 161, 175, 199$; and for $10000 \le n < 11000$, to exactly $954$ ones.

Below is the picture as above but for $n=123456789$:

Alternative question: Is it true that the set of $n$ for which the above phenomenon happens has natural density one? Is it cofinite? When it happens, does it involves the same constant $\alpha$?

There are exactly $1535$ numbers $n<10000$ for which the above phenomenon does not happen. The next picture displays for such $n$ the minimal $m$ such that $f^{m}(n) = f^{m-r}(n)$ for some $r>0$:

If the pattern continues like above up to infinity, they must have infinity many such $n$.
Below is their counting function (it looks logarithmic):