Invariance property for extended means

TL;DR

This paper investigates the invariance properties of extended mean-type mappings, linking the existence of unique invariant means to the ergodicity of an associated directed graph under certain conditions.

Contribution

It establishes a connection between invariant means of complex mean-type mappings and the ergodic properties of a related directed graph, providing a new approach to analyze their invariance.

Findings

Existence of unique invariant means reduces to graph ergodicity.

Provides conditions under which the invariant mean is unique.

Connects mean invariance to graph theoretical properties.

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 $\{(\alpha_{i,j},i) \colon i,j \text{ admissible}\}$.