For $r > 0$, define $f(n) = \lfloor {nr}\rfloor$ if $n$ is odd and $f(n) = \lfloor {n/r}\rfloor$ if $n$ is even. For which real numbers $r$ is the set $\{n,f(n), f(f(n)),\dots\}$ bounded for every nonnegative integer $n$?

So far, I have only computer-generated evidence. If $r = \sqrt 3$, the iterates reach $1$ for $n = 1,...,10^5$. If $r = \sqrt 5$, the iterates reach $0$ for $n = 1,...,10^5$. If $r = \sqrt 2$ and $n = 73$, the iterates appear to be unbounded. If $r = (1+\sqrt 5)/2$, it appears that the iterates are bounded if and only if $|n+1-2F| \le 1$ for some Fibonacci number $F$; the sequence of such $n$ begins with $0,1,2,3,4,5,6,8,9,10,14,15,16$.

A few more examples, in response to comments: if $r = 2^{1/3}$, the iterates for $n=9$ and $10 < n < 200001$ reach the cycle ${9,11,13,16,12}$. If $r = 2^{2/3}$, the iterates reach 1 for $n = 1,...,10^5$. For $r = e$ and $r = \pi$, the iterates reach $0$ for $n = 1,...,20000$. For $r = 1/9$, the iterates appear unbounded for many choices of $n$, whereas for $r = 2/9$, perhaps they are all bounded.