Starting with the fascinating question of Sebastien Palcoux Unexpected behavior involving √2 and parity let now $\alpha \not= \sqrt 2$ and consider the map
$$f: n \mapsto \left\{ \begin{array}{ll} \left \lfloor{n/\alpha} \right \rfloor \text{ if } n \text{ even,} \\ \left \lfloor{n \cdot \alpha} \right \rfloor \text{ if } n \text{ odd.} \end{array} \right.$$ As in the question of Palcoux let $f^{k+1} := f \circ f^k$ and consider the sequence $(f^k(73) \colon k \in \mathbb{N})$. As noted in the comments to Palcoux questions the sequence with $\alpha = \sqrt 2$ does not seem to be bounded. I tried some randomly chosen $\alpha = \sqrt {1.9}, \sqrt 3, \sqrt {2.1}, \ldots$. In all these cases the sequence was bounded and ended sometimes with $1$.
Is there any reason why the case $\alpha = \sqrt 2$ is so special?
