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We consider the asymptotic behavior of the eigenvalues of Toeplitz matriceswith rational symbol as the size of the matrix goes to infinity. Our mainresult is that the weak limit of the normalized eigenvalue counting measure isa particular component of the unique solution to a vector equilibrium problem.Moreover, we show that the other components describe the limiting behavior ofcertain generalized eigenvalues. In this way, we generalize the recent resultsof Duits and Kuijlaars for banded Toeplitz matrices.

An equilibrium problem for the limiting eigenvalue distribution of rational Toeplitz matrices