  There is a vast theory of the asymptotic behavior of orthogonal polynomials
with respect to a measure on $\mathbb{R}$ and its applications to Jacobi
matrices. That theory has an obvious affine invariance and a very special role
for $\infty$. We extend aspects of this theory in the setting of rational
functions with poles on $\overline{\mathbb{R}} = \mathbb{R} \cup \{\infty\}$,
obtaining a formulation which allows multiple poles and proving an invariance
with respect to $\overline{\mathbb{R}}$-preserving M\"obius transformations. We
obtain a characterization of Stahl--Totik regularity of a GMP matrix in terms
of its matrix elements; as an application, we give a proof of a conjecture of
Simon -- a Ces\`aro--Nevai property of regular Jacobi matrices on finite gap
sets.
