  In this paper we investigate the Brolin's theorem over $\mathbb{H}$, the skew
field of 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 Lyapunov 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.
