We will discuss the Brolin’s theorem over the quaternions. Moreover, considering a quaternionic polynomial p with real coefficients, we focus on the properties of its equilibrium measure, among the others, the mixing property and the Lya-punov exponents of the measure. We prove a central limit theorem and we compute the topological entropy and measurable entropy with respect to the quaternionic equilibrium measure. We prove that they are equal considering both a quaternionic polynomial with real coefficients and a polynomial with coefficients in a slice but not all real. Brolin’s theorems for the one slice preserving polynomials and for generic polynomials are also proved.