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Discrete planes, 2 -actions, Jacobi-Perron algorithm and substitutions

Pierre Arnoux, Valérie Berthé, Shunji Ito (2002)

Annales de l’institut Fourier

We introduce two-dimensional substitutions generating two-dimensional sequences related to discrete approximations of irrational planes. These two-dimensional substitutions are produced by the classical Jacobi-Perron continued fraction algorithm, by the way of induction of a 2 -action by rotations on the circle. This gives a new geometric interpretation of the Jacobi-Perron algorithm, as a map operating on the parameter space of 2 -actions by rotations.

Limit points of eigenvalues of truncated unbounded tridiagonal operators

E.K. Ifantis, C.G. Kokologiannaki, E. Petropoulou (2007)

Open Mathematics

Let T be a self-adjoint tridiagonal operator in a Hilbert space H with the orthonormal basis {e n}n=1∞, σ(T) be the spectrum of T and Λ(T) be the set of all the limit points of eigenvalues of the truncated operator T N. We give sufficient conditions such that the spectrum of T is discrete and σ(T) = Λ(T) and we connect this problem with an old problem in analysis.

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