Singular continuous spectrum and quantitative rates of weak mixing

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January 1998, 4(1): 33-42. doi: 10.3934/dcds.1998.4.33

Singular continuous spectrum and quantitative rates of weak mixing

1.	Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA 91125, United States

Received February 1996 Revised April 1997 Published October 1997

Abstract
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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.

Mathematics Subject Classification: Primary: 28D05, 28D20, 58F1.

Citation: Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33

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