Multi-variable translation equation which arises from homothety

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 ${(1-z)\phi({\bf x})=\phi(\phi({\bf x}z)(1-z)/z)}$; 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 when ${\phi}$ 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 ${\phi}$.