Turbulence phenomena in elementary real analysis.

Abstract

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.