Invariance Property for Extended Means

Abstract

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