Invariant means and iterates of mean-type mappings

Abstract

Classical result states that for two continuous and strict means $M,N:I^2\to I$ ($I$ is an interval) there exists a unique $(M,N)$-invariant mean $K:I^2\to I$, i.e. such a mean that $K\circ (M,N)=K$ and, moreover, the sequence of iterates $((M,N{)}^n{)}_{n=1}^{\infty }$ converge to $(K,K)$ pointwise. Recently it was proved that continuity assumption cannot be omitted in general. We show that if $K$ is a unique $(M,N)$-invariant mean then, under no continuity assumption, $(M,N{)}^n\to (K,K)$.