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# Transformations and Colorings of Groups

Published:2007-12-01
Printed: Dec 2007
• Yevhen Zelenyuk
• Yuliya Zelenyuk
 Format: HTML LaTeX MathJax PDF PostScript

## Abstract

Let $G$ be a compact topological group and let $f\colon G\to G$ be a continuous transformation of $G$. Define $f^*\colon G\to G$ by $f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume that $H=\Imag f^*$ is a subgroup of $G$ and for every measurable $C\subseteq H$, $\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable $C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that $f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$.
 Keywords: compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion
 MSC Classifications: 05D10 - Ramsey theory [See also 05C55] 20D60 - Arithmetic and combinatorial problems 22A10 - Analysis on general topological groups

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