Mahler Measure and its behavior under iteration

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 $\mathit{orbit}$ $\mathit{size}$ of $\alpha$, denoted $\# \mathcal{O}_M(\alpha)$, is the cardinality of the forward orbit of $\alpha$ under $M$. In this talk, we will start by introducing the definition of Mahler measure, briefly discuss results on the orbit sizes of algebraic numbers with degree at least 3 and non-unit norm, then we will turn our focus to the behavior of algebraic units, which are of interest in Lehmer's problem. We will mention the results regarding algebraic units of degree 4 and discuss that if $\alpha$ is an algebraic unit of degree $d\geq 5$ such that the Galois group of the Galois closure of $\mathbb{Q}(\alpha)$ contains $A_d$, then the orbit size must be 1, 2 or $\infty$. Furthermore, we will show that there exists units with orbit size larger than 2! This is joint work with Paul Fili and Lucas Pottmeyer.

number theory

Audience: researchers in the discipline

POINT: New Developments in Number Theory

Series comments: There will be two 20-minute contributed talks during each meeting aimed at a general number theory audience, including graduate students.

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