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On closed subgroups associated with involutions. (English) Zbl 1254.39010
Summary: Given an involution \(f\) on \((0,\infty)\), we prove that the set \(\mathcal C(f):= \{\lambda>0: \lambda f\) is an involution\(\}\) is a closed multiplicative subgroup of \((0,\infty)\) and therefore \(\mathcal C(f)\) is \(\{1\}\), or \(\lambda^{\mathbb Z}=\{\lambda^n: n\in\mathbb Z\}\) for some \(\lambda>0\), \(\lambda\neq 1\). Moreover, we provide examples of involutions possessing each one of the above types as the set \(\mathcal C(f)\) and prove that the unique involutions \(f\) such that \(\mathcal C(f)=(0,\infty)\) are \(f(x)=c/x\), \(c>0\).
MSC:
39B22 Functional equations for real functions
26A18 Iteration of real functions in one variable
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