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Cantor spectrum for Schrödinger operators with potentials arising from generalized skew-shifts. (English) Zbl 1165.37012

Continuous SL\((2,\mathbb R)\)- cocycles over a strictly ergodic homemorphism that fibers over an almost dynamical system showed (generalized skew-shifts) is considered in this work. It is proved that any cocycle that is not uniformly hyperbolic can be approximated by one that conjugate to an SO\((2,\mathbb R)\)-cocycle. A cocycle’s homotopy class does not display a certain obstruction to uniform hyperbolicity showed. For cocycles arising from Schrödinger operators it is concluded that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schrödinger operator is a Cantor set.
The basic results are presented in 12 lemmas, 10 remarks and 8 theorems.

MSC:

37D99 Dynamical systems with hyperbolic behavior
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
47B80 Random linear operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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