On closed subgroups associated with involutions. (English) Zbl 1254.39010

Cánovas, J. S.; Linero Bas, A.; Soler López, G.

Real Anal. Exch. 33(2007-2008), No. 2, 395-404 (2008).

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

Keywords:

involutions; closed subgroups; difference and functional equations