  We study the orthogonal polynomials associated with the equilibrium measure,
in logarithmic potential theory, living on the attractor of an Iterated
Function System. We construct sequences of discrete measures, that converge
weakly to the equilibrium measure, and we compute their Jacobi matrices via
standard procedures, suitably enhanced for the scope. Numerical estimates of
the convergence rate to the limit Jacobi matrix are provided, that show
stability and efficiency of the whole procedure. As a secondary result, we also
compute Jacobi matrices of equilibrium measures on finite sets of intervals,
and of balanced measures of Iterated Function Systems.
  These algorithms can reach large orders: we study the asymptotic behavior of
the orthogonal polynomials and we show that they can be used to efficiently
compute Green's functions and conformal mappings of interest in constructive
function theory.
