Has anyone studied the real function $$ f(x) = \frac{ 2 + 7x - ( 2 + 5x )\cos{\pi x}}{4}$$ (and $f(f(x))$ and $f(f(f(x)))$ and so on) with respect to the Collatz conjecture?
It does what Collatz does on integers, and is defined smoothly on all the reals.
I looked at $$\frac{ \overbrace{ f(f(\cdots(f(x)))) }^{\text{$n$ times}} }{x}$$ briefly, and it appears to have bounds independent of $n$.
Of course, the function is very wiggly, so Mathematica's graph is probably not $100\%$ accurate.
The analogue of the Collatz conjecture would be false, since the image of an interval of length $1$, say, contains an interval of length $1$ strictly to the right. That means you can find a point whose images keep moving to the right.