Spectra of non-commutative dynamical systems and graphs related to fractal groups

@article{Bartholdi2000SpectraON,
  title={Spectra of non-commutative dynamical systems and graphs related to fractal groups},
  author={Laurent Bartholdi and Rostislav I. Grigorchuk},
  journal={Comptes Rendus Mathematique},
  year={2000},
  volume={331},
  pages={429-434}
}
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References

SHOWING 1-5 OF 5 REFERENCES
Répartition asymptotique des valeurs propres de l’opérateur de Hecke _
La répartition asymptotique des valeurs propres des opérateurs de Hecke Tp, pour p premier variable, est un problème intéressant et difficile, sur lequel on ne dispose que de résultats partiels, cf.
Spectra of non-commutative dynamical systems and graphs related to fractal groups
  • Comptes rendus mathématique,
  • 2000