On the behavior of Mahler's measure under iteration

TL;DR

This paper investigates the dynamical behavior of Mahler's measure as a self-map on algebraic numbers, revealing diverse orbit sizes and specific constraints for algebraic units of certain degrees.

Contribution

It establishes the possible orbit sizes for algebraic numbers under Mahler's measure, including existence results for all sizes and degree-specific restrictions.

Findings

Existence of algebraic numbers with every orbit size for degrees ≥ 3

Algebraic units of degree 4 have orbit sizes only 1, 2, or infinity

Existence of algebraic units with arbitrarily large finite orbit sizes

Abstract

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.