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DCDS

We prove that for a dense $G_{\delta}$ of shift-invariant measures
on $A^{\ZZ^d}$, all $d$ shifts have purely singular continuous spectrum
and give a new proof that in the weak topology of measure preserving $\ZZ^d$
transformations, a dense $G_{\delta}$ is generated by transformations
with purely singular continuous spectrum.
We also give new examples of smooth unitary cocycles over an
irrational rotation which have purely singular continuous spectrum.
Quantitative weak mixing properties are related by results of Strichartz and
Last to spectral properties of the unitary Koopman operators.

keywords:
spectral theory
,
invariant measures
,
singular continuous spectrum.
,
mixing
,
Ergodic theory

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