  In this paper, we present a new method for the analysis of piecewise
dynamical systems that are similar to the Collatz conjecture in regard to
certain properties of the commutator of their sub-functions. We use the fact
that the commutator of polynomials $E(n)=n/2$ and $O(n)=(3n+1)/2$ is constant
to study rearrangements of compositions of $E(n)$ and $O(n)$. Our main result
is that for any positive rational number $n$, if $(E^{e_1} \circ O^{o_1} \circ
E^{e_2} \circ \dotsb \circ O^{o_l} \circ E^{e_{l+1}})(n)=1$, then $(E^{e_1}
\circ O^{o_1} \circ E^{e_2} \circ \dotsb \circ O^{o_l} \circ E^{e_{l+1}})(n) =
\lceil(E^{e_1 + \dotsb + e_{l+1}} \circ O^{o_1 + \dotsb + o_{l}})(n)\rceil$,
where exponentiation is used to denote repeated composition and $e_i$ and $o_i$
are positive integers. Composition sequences of this form have significance in
the context of the Collatz conjecture. The techniques used to derive this
result can be used to produce similar results for a wide variety of repeatedly
composed piecewise functions.
