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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Lévy group action and invariant measures on $ \beta \mathbb {N}$

Author: Martin Blümlinger
Journal: Trans. Amer. Math. Soc. 348 (1996), 5087-5111
MSC (1991): Primary 54H20
MathSciNet review: 1390970
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Abstract: For $f\in \ell ^{\infty }( \mathbb {N})$ let $Tf$ be defined by $Tf(n)=\frac {1}{n}\sum _{i=1}^{n}f(i)$. We investigate permutations $g$ of $ \mathbb {N}$, which satisfy $Tf(n)-Tf_{g}(n)\to 0$ as $n\to \infty $ with $f_{g}(n)=f(gn)$ for $f\in \ell ^{\infty }( \mathbb {N})$ (i.e. $g$ is in the Lévy group $ \mathcal {G})$, or for $f$ in the subspace of Cesàro-summable sequences. Our main interest are $ \mathcal {G}$-invariant means on $\ell ^{\infty }( \mathbb {N})$ or equivalently $ \mathcal {G}$-invariant probability measures on $ \beta \mathbb {N}$. We show that the adjoint $T^{*}$ of $T$ maps measures supported in $ \beta \mathbb {N} \setminus \mathbb {N}$ onto a weak*-dense subset of the space of $ \mathcal {G}$-invariant measures. We investigate the dynamical system $( \mathcal {G}, \beta \mathbb {N})$ and show that the support set of invariant measures on $ \beta \mathbb {N}$ is the closure of the set of almost periodic points and the set of non-topologically transitive points in $ \beta \mathbb {N}\setminus \mathbb {N}$. Finally we consider measures which are invariant under $T^{*}$.

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Martin Blümlinger
Affiliation: Institut 114, Technische Universität Wien, Wiedner Hauptstr. 8-10, 1040 Wien, Austria

Received by editor(s): September 29, 1995
Additional Notes: Part of this work was carried out at Macquarie University with financial support from the Australian Research Council.
Article copyright: © Copyright 1996 American Mathematical Society