Involutions of real intervals (Q2875405)

A real (non-trivial) involution is a continuous map \(h\) defined from a real subinterval \(J\) into itself (being \(h\) distinct to the identity map \(\mathrm{Id}|_{J}\)) such that its second iterate is equal to \(\mathrm{Id}_J\). It is not restrictive to assume that \(0\in J.\) In the first sections, the author presents two new results about this class of maps, illustrated by suitable examples.NEWLINENEWLINEThe first one establishes a new method to construct convolutions from even functions defined on symmetric intervals, namely if \(\varphi:I\rightarrow \mathbb R\) is a continuous even map, with \(\varphi(0)=0\) and \(I=(-a,a)\) a non-degenerate subinterval, and we know that the map \(K:I\rightarrow J\) defined by \(K(y)=\frac{1}{2}\left(y+\varphi(y)\right)\) is a homeomorphism onto some \(J\), then the map \(h(x):=x-K^{-1}(x)\) is a new involution on \(J\).NEWLINENEWLINEThe second result asserts that if \(f=f(x,y)\) is a real-valued \(C^1\) function defined on an open set \(\Omega\subset\mathbb R^2\), with \((0,0)\in\Omega\) and \(f(0,0)=0\), and such that for any point \((x,y)\in\Omega\) we have \((y,x)\in\Omega\) and \(f(x,y)=f(x,y)\), then the connected component \(\Gamma\) of \(f^{-1}(0)\) that contains the origin is the graph of a smooth involution \(h\), whenever \(\frac{\partial f}{\partial y}\neq 0\) in \(\Gamma\); even more, all smooth involutions are obtained in the same manner.NEWLINENEWLINEIn the last part, the author gives a little account on the application of (smooth) involutions in the setting of continuous dynamical systems. They are used: a) to characterize isochronous potentials for the scalar differential equation \(\ddot{x}=-g(x)\), see [\textit{G. Zampieri}, J. Differ. Equations 78, No. 1, 74--88 (1989; Zbl 0701.34049)]; b) to analyze the Lyapunov stability around the origin of a certain differential system representing the motion under attractive central forces, see [\textit{G. Zampieri}, J. Differ. Equations 74, No. 2, 254--265 (1988; Zbl 0668.34051)]; c) to solve explicitly a certain functional-differential equation involving the involution \(h(x)=-\frac{x}{1+x}\), \(x >1\).