Cantor singular continuous spectrum for operators along interval exchange transformations

Cobo, Milton; Gutiérrez, Carlos; Oliveira, César R. de

doi:10.1090/s0002-9939-07-09074-0

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“…Interval Exchange Transformations. Recall that an IET subshift is determined by an irreducible permutation τ and a probability vector λ. Cobo, Gutierrez, and de Oliveira showed the following result in [35] (see also [73]). Theorem 4.25.…”

Section: 4mentioning

confidence: 83%

Spectral Properties of Schrödinger Operators Arising in the Study of Quasicrystals

Damanik

Embree

Gorodetski

2015

Progress in Mathematics

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We consider discrete Schrödinger operators with pattern Sturmian potentials. This class of potentials strictly contains the class of Sturmian potentials, for which the spectral properties of the associated Schrödinger operators are well understood. In particular, it is known that for every Sturmian potential, the associated Schrödinger operator has zero-measure spectrum and purely singular continuous spectral measures. We conjecture that the same statements hold in the more general class of pattern Sturmian potentials. We prove partial results in support of this conjecture. In particular, we confirm the conjecture for all pattern Sturmian potentials that belong to the family of Toeplitz sequences. Date

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Section: 4mentioning

confidence: 83%