Generic spectral properties of measure-preserving maps and applications

Authors:
Andrés del Junco and Mariusz Lemańczyk

Journal:
Proc. Amer. Math. Soc. **115** (1992), 725-736

MSC:
Primary 28D05; Secondary 47A35

DOI:
https://doi.org/10.1090/S0002-9939-1992-1079889-7

MathSciNet review:
1079889

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Abstract: Let denote the group of all automorphisms of a finite Lebesgue space equipped with the weak topology. For , let denote its maximal spectral type.

**Theorem 1.** *There is a dense* *subset* *such that, for each* *and* , *the convolutions*

*are mutually singular, provided that*( )

*is not a rearrangement of*.

Theorem 1 has the following consequence.

**Theorem 2.** *has a dense* *subset* *such that for* *the following holds: For any* *and* , *the only way that* , *or any factor thereof, can sit as a factor in* is inside the *th coordinate* *-algebra for some* *with* .

Theorem 2 has applications to the construction of certain counterexamples, in particular nondisjoint automorphisms having no common factors and weakly isomorphic automorphisms that are not isomorphic.

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1079889-7

Article copyright:
© Copyright 1992
American Mathematical Society