  For an algebraic number $\alpha$ we denote by $M(\alpha)$ the Mahler measure
of $\alpha$. As $M(\alpha)$ is again an algebraic number (indeed, an algebraic
integer), $M(\cdot)$ is a self-map on $\overline{\mathbb{Q}}$, and therefore
defines a dynamical system. The \emph{orbit size} of $\alpha$, denoted $\#
\mathcal{O}_M(\alpha)$, is the cardinality of the forward orbit of $\alpha$
under $M$. We prove that for every degree at least 3 and every non-unit norm,
there exist algebraic numbers of every orbit size. We then prove that for
algebraic units of degree 4, the orbit size must be 1, 2, or infinity. We also
show that there exist algebraic units of larger degree with arbitrarily large
but finite orbit size.
