No-wandering-intervals conjecture for the real $3x+1$ extension

Let $f$ be the real analytic extension of the $3x+1$ map on $\mathbb{R}^{+}$.

No-wandering-intervals conjecture. The function $f$ has no wandering intervals in $\mathbb{R}^{+}$.

The paper describes the existence of wandering intervals as an open question with important implications for the $3x+1$ problem, and proves logical implications among this conjecture and the unbounded-orbit and discrete formulations.