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2003-2004 Turbulence phenomena in elementary real analysis.
Nikolaos Efstathiou Sofronidis
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Real Anal. Exchange 29(2): 813-820 (2003-2004).


The purpose of this note is to show that if $-\infty < \alpha < \beta < \infty $ and $E_{\alpha }^{\beta }$ is the equivalence relation, which is defined on the Polish group $C([\alpha ,\beta ), {\mathbb R}_{+}^{*})$ by $fE_{\alpha }^{\beta }g \iff \lim_{x\rightarrow {\beta }^{-}}\frac{f(x)}{g(x)}=1$, where $f$, $g$ are in $C([\alpha ,\beta ),{\mathbb R}_{+}^{*})$, then $E_{\alpha }^{\beta }$ is induced by a turbulent Polish group action. Hence if $L$ is any countable language and ${\mathcal{A}}:C([\alpha ,\beta ),{\mathbb R}_{+}^{*})\rightarrow X_{L}$ is any Baire measurable function from the Polish group $C([\alpha ,\beta ), {\mathbb R}_{+}^{*})$ to the Polish space $X_{L}$ of countably infinite structures for $L$ with the property that $fE_{\alpha }^{\beta }g \Rightarrow {\mathcal{A}}(f) \cong {\mathcal{A}}(g)$, whenever $f$, $g$ are in $C([\alpha ,\beta ),{\mathbb R}_{+}^{*})$, then there exists a $E_{\alpha }^{\beta }$-invariant comeager subset $S$ of $C([\alpha ,\beta ),{\mathbb R}_{+}^{*})$ for which all countable structures in ${\mathcal{A}}[S]$ are isomorphic.


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Nikolaos Efstathiou Sofronidis. "Turbulence phenomena in elementary real analysis.." Real Anal. Exchange 29 (2) 813 - 820, 2003-2004.


Published: 2003-2004
First available in Project Euclid: 7 June 2006

zbMATH: 1065.03030
MathSciNet: MR2083816

Primary: 03E15 , 26A99

Keywords: Asymptotic Equality , turbulence

Rights: Copyright © 2003 Michigan State University Press

Vol.29 • No. 2 • 2003-2004
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