A generalisation of Mahler measure and its application in algebraic dynamical systems

We prove a generalisation of the entropy formula for certain algebraic ${\displaystyle {\mathbb{Z}}^d}$-actions given in [2] and [4]. This formula expresses the entropy as the logarithm of the Mahler measure of a Laurent polynomial in d variables with integral coefficients. We replace the rational integers by the integers in a number field and examine the entropy of the corresponding dynamical system.