Mean-type mappings and invariance principle

TL;DR

This paper explores mean-type mappings in finite dimensions, establishing conditions for convergence of orbits and invariance, and demonstrates their application in solving functional equations.

Contribution

It introduces new results linking the uniqueness of invariant means to orbit convergence and constructs a strongly irregular mean-type mapping with a unique invariant mean.

Findings

Uniqueness of invariant mean implies orbit convergence.

Constructed a strongly irregular mean-type mapping.

Applied findings to solve a functional equation.

Abstract

In the finite dimensional case, mean-type mappings, their invariant means, relations between the uniqueness of invariant means and convergence of orbits of the mapping, are considered. In particular it is shown, that the uniqueness of an invariance mean implies the convergence of all orbits. A strongly irregular mean-type mapping is constructed and its unique invariant mean is determined. An application in solving a functional equation is presented.