I am studying a problem which I think has a striking resemblance to the Collatz conjecture. Essentially, given a natural number $k_0$, there is an algorithm which produces another natural number $k_1$ and this process repeats, giving us a trajectory for $k_n$, essentially a dynamical system.
The Algorithm
Fix real numbers $\omega_1, \omega_2 > 0$. Given $k > 0$ integer, find an integer $n$ such that
\begin{equation} k = n+ \lfloor \tfrac{2n-1+\omega_1}{2 \omega_1}\rfloor \end{equation} then replace $\omega_1$ with $\omega_2$ to get the next iteration.
If no such $n$ exists, then find an integer $m$ such that \begin{equation} k= m + \lfloor \omega_1 m - \tfrac{\omega_1 }{2} +\tfrac12 \rfloor \end{equation} then replace $\omega_1$ with $\omega_2$ to get the next iteration.
It turns out that these two are mutually exclusive: you can always find an $m$ if you can't find an $n$, and vice versa. You can't find $n,m$ which simultaneously works.
Simulating a trajectory for a very large values of $k$, it appears that there is a behaviour similar to a (geometric) random walk. In the literature, it seems to me that something similar happens in the collatz problem, and there are stochastic models as a random walk.
$k$, the trajectory appears to resemble a random walk." />
My question is the following: I am interested in if there is exponential behaviour of the $k$ values, and how `common' this is.
I was wondering if it is possible to approximate this dynamical system by a stochastic model? How could I check if it is valid to think of this process as a random walk? In particular, how could I check that this system has a Markovian property? I was furthermore interested in if any of the work done by Tao in https://arxiv.org/pdf/1909.03562.pdf would be valid here or transferable because he appears to explore almost the same question. I am not too familiar with this area of mathematics so I would tremendously appreciate any ideas on literature to explore.