Means of iterates

TL;DR

This paper characterizes certain continuous bijections on real intervals whose iterates relate to quasi-arithmetic means, showing they are composed of at most three affine segments under specific parity conditions.

Contribution

It provides a classification of functions with iterates forming quasi-arithmetic means, revealing their piecewise affine structure under parity constraints.

Findings

Functions are at most three affine pieces when at most one of k,n is odd.

The study links iterates of functions to quasi-arithmetic means.

Provides a structural characterization of such functions.

Abstract

We determine continuous bijections $f$, acting on a real interval into itself, whose $k$-fold iterate is the quasi-arithmetic mean of all its subsequent iterates from $f^0$ up to $f^n$ (where $0\le k\le n$). Namely, we prove that if at most one of the numbers $k,n$ is odd, then such functions consist of at most three affine pieces.