Multi-variable translation equation which arises from homothety

Details Multi-variable translation equation which arises from homothety Multi-variable translation equation which arises from homothety

2009-11-08

arxiv.org/abs/0911.1513v3

Aequationes Math. 80 (3) (2010), 335-350

Mathematics

Classical Analysis and ODEs

15 pages

Scientific paper

10.1007/s00010-010-0032-3

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.

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