 95193 Knill O.
 Singular Continuous Spectrum in Ergodic Theory
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Apr 11, 95

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Abstract. We prove that in the weak topology of measure preserving transformations,
a dense $G_{\delta}$
has purely singular continuous spectrum in the orthocomplement of the
constant functions. In the uniform topology, a dense $G_{\delta}$
of aperiodic transformations has singular continuous spectrum.
We show that a dense $G_{\delta}$ of shiftinvariant
measures has purely singular continuous spectrum. These results stay
true for $\ZZ^d$ actions of measure preserving transformations.
There exist smooth unitary cocycles over an
irrational rotation which have purely singular continuous spectrum.
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