Knill O.
Singular Continuous Spectrum in Ergodic Theory
(30K, LaTeX)
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 shift-invariant
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.