Absence of nontrivial cycles for the $3x+1$ map

Let $T$ be the function on positive integers defined by

$$T(n)=\begin{cases}(3n+1)/2&\text{if }n\text{ is odd},\\ n/2&\text{otherwise.}\end{cases}$$

Absence of nontrivial cycles. There do not exist integers $n>2$ and $k>0$ such that $T^k(n)=n$.

Together with the absence of divergent trajectories, this is presented as an equivalent dynamical formulation of the $3x+1$ problem and remains open.