Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences

Blümlinger, M.; Obata, N.

Annales de l'Institut Fourier, Tome 41 (1991) no. 3, pp. 665-678.

Nous considérons les permutations de $\mathbf{N}$ qui conservent la $\mu$-répartition des suites ou la densité des parties de $\mathbf{N}$ ou la somme de Cesàro des suites sommables, et montrons que le groupe (resp. semi-groupe) de ces permutations sont les mêmes. Il est prouvé qu’il y a des fonctionnelles de $l^{\infty }\left(\mathbf{N}\right)$ qui sont invariantes sous l’action du groupe de Lévy et que toutes ces fonctionnelles sont des extensions de la somme de Cesàro.

We are interested in permutations preserving certain distribution properties of sequences. In particular we consider $\mu$-uniformly distributed sequences on a compact metric space $X$, 0-1 sequences with densities, and Cesàro summable bounded sequences. It is shown that the maximal subgroups, respectively subsemigroups, of $Aut\left(\mathbf{N}\right)$ leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group $𝒢$. We show that $𝒢$ is big in the sense that the Cesàro mean is characterized by its invariance under the Lévy group. As a result, any $𝒢$ -invariant positive normalized linear functional on $l^{\infty }\left(\mathbf{N}\right)$ is an extension of Cesàro means. Finally we prove that there exist $𝒢$ -invariant extensions of Cesàro mean to all of $l^{\infty }\left(\mathbf{N}\right)$.

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