Lévy group action and invariant measures on 𝛽{ℕ}

Abstract

For f ∈ ℓ ∞ ( N ) f\in \ell ^{\infty }( \mathbb {N}) let T f Tf be defined by T f ( n ) = 1 n ∑ i = 1 n f ( i ) Tf(n)=\frac {1}{n}\sum _{i=1}^{n}f(i) . We investigate permutations g g of N \mathbb {N} , which satisfy T f ( n ) − T f g ( n ) → 0 Tf(n)-Tf_{g}(n)\to 0 as n → ∞ n\to \infty with f g ( n ) = f ( g n ) f_{g}(n)=f(gn) for f ∈ ℓ ∞ ( N ) f\in \ell ^{\infty }( \mathbb {N}) (i.e. g g is in the Lévy group G ) \mathcal {G}) , or for f f in the subspace of Cesàro-summable sequences. Our main interest are G \mathcal {G} -invariant means on ℓ ∞ ( N ) \ell ^{\infty }( \mathbb {N}) or equivalently G \mathcal {G} -invariant probability measures on β N \beta \mathbb {N} . We show that the adjoint T ∗ T^{*} of T T maps measures supported in β N ∖ N \beta \mathbb {N} \setminus \mathbb {N} onto a weak*-dense subset of the space of G \mathcal {G} -invariant measures. We investigate the dynamical system ( G , β N ) ( \mathcal {G}, \beta \mathbb {N}) and show that the support set of invariant measures on β N \beta \mathbb {N} is the closure of the set of almost periodic points and the set of non-topologically transitive points in β N ∖ N \beta \mathbb {N}\setminus \mathbb {N} . Finally we consider measures which are invariant under T ∗ T^{*} .