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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
MathSciNet review: 1079889
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Abstract: Let $ \mathcal{K}$ denote the group of all automorphisms of a finite Lebesgue space equipped with the weak topology. For $ T \in \mathcal{K}$, let $ {\sigma _T}$ denote its maximal spectral type.

Theorem 1. There is a dense $ {G_\delta }$ subset $ G \subset \mathcal{K}$ such that, for each $ T \in G$ and $ k(1), \ldots ,k(l) \in {\mathbb{Z}^ + },k'(1), \ldots ,k'(l') \in {\mathbb{Z}^ + }$, the convolutions

$\displaystyle {\sigma _{{T^{k(1)}}}}* \cdots *{\sigma _{{T^{k(l)}}}}\quad and\quad {\sigma _{{T^{k'(1)}}}}* \cdots *{\sigma _{{T^{k'(l')}}}}$

are mutually singular, provided that ( $ (k(1), \ldots ,k(l))$) is not a rearrangement of $ (k'(1), \ldots ,k'(l'))$.

Theorem 1 has the following consequence.

Theorem 2. $ \mathcal{K}$ has a dense $ {G_\delta }$ subset $ F \subset G$ such that for $ T \in F$ the following holds: For any $ {\mathbf{k}}:\mathbb{N} \to \mathbb{Z} - \{ 0\} $ and $ l \in \mathbb{Z} - \{ 0\} $, the only way that $ {T^l}$, or any factor thereof, can sit as a factor in $ {T^{{\mathbf{k}}(1)}} \times {T^{{\mathbf{k}}(2)}} \times \cdots $ is inside the $ i$th coordinate $ \sigma $-algebra for some $ i$ with $ {\mathbf{k}}(i) = l$.

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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Article copyright: © Copyright 1992 American Mathematical Society