Canonical decomposition in ${\mathbb{r}}_{+}^{\ast }$ of a convergent natural number by the collatz iterations

The Collatz variations pattern seems not to have any recurrence relation between numbers. But knowing that there is at least a natural number that converges after several iterations we construct a function $f_{X,Y}$ that is equal to the value of convergence for all convergent sequences. A canonical decomposition can be expressed for such numbers.