  We study the properties and asymptotics of the Jacobi matrices associated
with equilibrium measures of the weakly equilibrium Cantor sets. These family
of Cantor sets were defined and different aspects of orthogonal polynomials on
them were studied recently. Our main aim is numerically examine some
conjectures concerning orthogonal polynomials which do not directly follow from
previous results. We also compare our results with more general conjectures
made for recurrence coefficients associated with fractal measures supported on
$\mathbb{R}$.
