Functions for the $$$3n+1$$$-problem

Abstract

What functions can represent the $$$3n+1$$$-problem? A number of such functions have been given during the past years. In this paper, a specific one is considered which has $$$1, 2, 3, \ldots$$$ as critical points and $$$1.5, 2.5, 3.5, \ldots$$$ as fixed points on the interval $$$[1,\infty)$$$. Another function is presented with critical points $$$1, 2, 3, \ldots ,$$$ and fixed point $$$0.5391\!\ldots $$$ on the real line.