On the behavior of Mahler\u2019s measure 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 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.