an:05499478
Zbl 1254.39010
C??novas, J. S.; Linero Bas, A.; Soler L??pez, G.
On closed subgroups associated with involutions
EN
Real Anal. Exch. 33(2007-2008), No. 2, 395-404 (2008).
00233472
2008
j
39B22 26A18
involutions; closed subgroups; difference and functional equations
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 \(f(x)=c/x\), \(c>0\).