One of the supporting arguments for the Collatz conjecture is the probability heuristic, which states roughly that because the collatz operations tends to decrease numbers over time, it probably doesn't diverge.
Are there examples of where this isn't true, i.e. is there a variant of the collatz conjecture for which the probability heuristic holds, but not all numbers converge to a cycle? (Preferably, the set of numbers that diverge should be a non-null set.)
Let $A$ be any infinite subset of $\mathbb{N}{\,\setminus}\{1\}$, with positive density less than $1/2$.
For $a \in A$, let $s(a)$ be the least element of $A$ which is greater than $a$.
Define $f:\mathbb{N}\to \mathbb{N}$ by $$ f(n)= \begin{cases} s(n)&\text{if}\;n\in A\\[4pt] 1&\text{otherwise}\\ \end{cases} $$ Then probabilistically, every iteration should cycle, but clearly, the iterations which start with an element of $A$, approach infinity.