  We present correspondences induced by some classical mappings between
measures on an interval and measures on the unit circle. More precisely, we
link their sequences of orthogonal polynomial and their recursion coefficients.
We also deduce some correspondences between particular equilibrium measures of
random matrix ensembles. Additionally, we show that these mappings open up
gateways between the sum rules associated with some classical models, leading
to new formulations of several sum rules.
