Multi-variable translation equation which arises from homothety

TL;DR

This paper fully solves a specific multi-variable translation functional equation arising from homothety, identifying four solutions in the continuous case and presenting a solution without regularity assumptions in one dimension.

Contribution

It provides a complete classification of solutions to the multi-variable translation equation in the continuous case and extends the analysis to a non-regular one-dimensional solution.

Findings

Exactly four solutions in the continuous case up to conjugation

Existence of a non-regular solution in one dimension

Complete characterization of solutions in the specified setting

Abstract

In many regular cases, there exists a (properly defined) limit of iterations of a function in several real variables, and this limit satisfies the functional equation (1-z)f(x)=f(f(xz)(1-z)/z); here z is a scalar and x is a vector. This is a special case of a well-known translation equation. In this paper we present a complete solution to this functional equation in case f is a continuous function on a single point compactification of a 2-dimensional real vector space. It appears that, up to conjugation by a homogeneous continuous function, there are exactly four solutions. Further, in a 1-dimensional case we present a solution with no regularity assumptions on f.