  We reformulate the $3x+1$ conjecture by restricting attention to numbers
congruent to $2$ (mod $3$). This leads to an equivalent conjecture for positive
integers that reveals new aspects of the dynamics of the $3x+1$ problem.
Advantages include a governing function with particularly simple mapping
properties in terms of partitions of the set of integers. We use the refined
conjecture to obtain a new characterization of $3x+1$ trajectories that shows a
special role played by numbers congruent to $2$ or $8$ (mod $9$). We construct
an accelerated iteration whose long-term behavior involves only those numbers.
