Functions for the $3n+1$-problem

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.