# Invariant means and iterates of mean-type mappings

## TL;DR

This paper proves that for mean-type mappings with a unique invariant mean, the iterates always converge to that mean, even without assuming continuity, extending classical results.

## Contribution

It generalizes the convergence of iterates of mean-type mappings to cases without continuity assumptions, given the uniqueness of the invariant mean.

## Key findings

- Iterates of mean-type mappings converge to the invariant mean.
- Uniqueness of the invariant mean guarantees convergence.
- Convergence holds without continuity assumptions.

## Abstract

Classical result states that for two continuous and strict means $M,\,N \colon I^2 \to I$ ($I$ is an interval) there exists a unique $(M,N)$-invariant mean $K \colon I^2 \to I$, 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, $(M,N)^n \to (K,K)$.