zbMATH — the first resource for mathematics

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
Full Text: