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

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 $T^{*}$.