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Transformations and Colorings of Groups

  Published:2007-12-01
 Printed: Dec 2007
  • Yevhen Zelenyuk
  • Yuliya Zelenyuk
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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 compact topological group, continuous transformation, endomorphism, Ramsey theoryinversion
MSC Classifications: 05D10, 20D60, 22A10 show english descriptions Ramsey theory [See also 05C55]
Arithmetic and combinatorial problems
Analysis on general topological groups
05D10 - Ramsey theory [See also 05C55]
20D60 - Arithmetic and combinatorial problems
22A10 - Analysis on general topological groups
 

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