Wandering points for the Mahler measure

TL;DR

This paper investigates the behavior of points under Mahler's measure within algebraic number fields, providing a complete classification for abelian fields and low-degree extensions.

Contribution

It offers a complete solution to the wandering points problem for abelian number fields and degree up to five extensions, advancing understanding of Mahler's measure dynamics.

Findings

Classified wandering points for all abelian number fields.

Extended results to degree five extensions.

Provided a comprehensive framework for Mahler measure dynamics.

Abstract

Mahler's measure defines a dynamical system on the algebraic numbers. In this paper, we study the problem of which number fields have points which wander under the iteration of Mahler's measure. We completely solve the problem for all abelian number fields, and more generally, for all extensions of the rationals of degree at most five.