| Tue, Oct 29, 2019
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Number Theory Seminar
3:30 PM
MSCS 422 | | On the Stopping Time of Algebraic Units under iteration of Mahler's Measure
Mingming Zhang, OSU
| | | 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 stopping time of $\alpha$, denoted $ST(\alpha)$, 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 stopping time. We then prove that for algebraic units of degree 4, the stopping time must be 1, 2, or $\infty$. Moreover, if $\alpha$ is an algebraic unit of degree $d\geq 5$ such that the Galois group of the Galois closure of $\mathbb{Q}$ contains $A_d$ then the stopping time must be 1. 2 or $\infty$. Finally, for every degree at least 12 and divisible by 6, there exist algebraic units of arbitrarily large stopping time. This is joint work with Paul Fili and Lucas Pottmeyer. |
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