Multi-variable translation equation which arises from homothety |
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Authors: | Giedrius Alkauskas |
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Institution: | 1. Department of Integrative Biology, Institute of Mathematics, Universit?t für Bodenkultur Wien, Gregor Mendel-Stra?e 33, 1180, Vienna, Austria 2. Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225, Vilnius, Lithuania
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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
_boxclose)=((xz)(1-z)/z){(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 f{\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 f{\phi}. |
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Keywords: | |
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