Involutions of real intervals

AbstractThis paper shows a simple construction of the continuous involu-tions of real intervals in terms of the continuous even functions. Wealso study the smooth involutions de ned by symmetric equations. Fi-nally, we review some applications, in particular the characterizationof the isochronous potentials by means of smooth involutions. 1 Introduction An involution is a function that is its own inverse. This is an important objectin all mathematical elds. We are going to consider continuous involutionson real intervals only.Proposition 1.1. Let h : J !J be continuous function on the intervalJR which is the inverse of itself and does not coincide with the identityfunction id J . Then h is strictly decreasing and has a unique xed pointx = h(x). To appear in Annales Polonici Mathematici.2010 Mathematics Subject Classi cation: Primary 39B22; Secondary 37J45.Key words and phrases: Involutions of real intervals; even functions; symmetric equa-tions; isochronous potentials. 1 arXiv:1309.4912v1 [math.CA] 19 Sep 2013