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.

中文翻译：

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马勒测度在迭代下的行为

对于代数数 $\alpha$，我们用 $M(\alpha)$ 表示 $\alpha$ 的马勒测度。由于 $M(\alpha)$ 又是一个代数数（确实是一个代数整数），$M(\cdot)$ 是 $\overline{\mathbb{Q}}$ 上的一个自映射，因此定义了一个动力系统。$\alpha$的\emph{轨道大小}，记为$\# \mathcal{O}_M(\alpha)$，是$\alpha$在$M$下的前向轨道的基数。我们证明，对于每个至少为 3 的度数和每个非单位范数，都存在每个轨道大小的代数数。然后我们证明，对于 4 次代数单位，轨道大小必须是 1、2 或无穷大。我们还表明，存在具有任意大但有限轨道尺寸的更大次数的代数单位。