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SUMMARY:Mingming Zhang (Oklahoma State University)
DTSTART:20201005T133000Z
DTEND:20201005T140000Z
DTSTAMP:20260423T021231Z
UID:POINT/14
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/POINT/14/">M
 ahler Measure and its behavior under iteration</a>\nby Mingming Zhang (Okl
 ahoma State University) as part of POINT: New Developments in Number Theor
 y\n\n\nAbstract\nFor 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 $\\o
 verline{\\mathbb{Q}}$\, and therefore defines a dynamical system. The $\\m
 athit{orbit}$ $\\mathit{size}$ 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 mea
 sure\,  briefly discuss results on the orbit sizes of algebraic numbers wi
 th degree at least 3 and non-unit norm\, then we will turn our focus to th
 e behavior of algebraic units\, which are of interest in Lehmer's problem.
  We will mention the results regarding algebraic units of degree 4 and dis
 cuss 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 j
 oint work with Paul Fili and Lucas Pottmeyer.\n
LOCATION:https://researchseminars.org/talk/POINT/14/
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