Asymptotic Behavior of Jacobi Matrices of IFS: A Long-Standing Conjecture

We study numerically the asymptotic behavior of Jacobi matrices of measures supported on finite sets of interval, for which a complete theory exists, and on Cantor sets, whose properties (almost periodicity [G. Mantica, Quantum intermittency in almost periodic systems derived from their spectral properties, Physica D 103 (1997) 576–589]) are still conjectural. We employ tools from logarithmic potential theory and iterated function systems. In the latter case, our results seem to be consistent with almost periodicity in the sense of Besicovitch.