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

Orthogonal rational functions with real poles, root asymptotics, and GMP matrices