Invariance property for extended means

Abstract

e study the properties of the mean-type mappings ${\bf M}\colon I^p \to I^p$ of the form $${\bf M}(x_1,\dots,x_p):=\big(M_1(x_{\alpha_{1,1}},\dots,x_{\alpha_{1,d_1}}),\dots,M_p(x_{\alpha_{p,1}},\dots,x_{\alpha_{p,d_p}})\big),$$ where $p$ and $d_i$-s are positive integers, each $M_i$ is a $d_i$-variable mean on an interval $I \subset \mathbb{R}$, and $\alpha_{i,j}$-s are elements from $\{1,\dots,p\}$. We show that, under some natural assumption on $M_i$-s, the problem of existing the unique $\bf M$-invariant mean can be reduced to the ergodicity of the directed graph with vertexes $\{1,\dots,p\}$ and edges $\{(\alpha_{i,j},i) \colon i,j \text{ admissible}\}$.