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Linearly recurrent circle map subshifts and an application to Schrödinger operators. (English) Zbl 1023.47019

The authors consider the discrete, one-dimensional Schrödinger operators with potentials generated by circle maps [D. Lenz, Commun. Math. Phys. 227, 119-130 (2002; Zbl 1065.47035)], where each potential contains two parameters. The set of these parameters are characterized so that the corresponding circle map is linearly recurrent [F. Durand, Ergodic Theory Dyn. Syst. 20, 1061-1078 (2000; Zbl 0965.37013)]. This property allows the authors to conclude that the discrete Schrödinger operator has a purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure.

MSC:

47B39 Linear difference operators
39A70 Difference operators
37E10 Dynamical systems involving maps of the circle
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
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