On Zhang-Yang's open question about iterative roots of PM functions of height 1 (II): Decreasing case

Abstract

A Zhang-Yang's open question reads: Does a PM function F with height $H\left(F\right)=1$ have an iterative root f of order $n\le N\left(F\right)+1$ if the ‘characteristic endpoints condition’ is not satisfied? This question was answered in the case that F is strictly increasing on its characteristic interval $K\left(F\right)$. However, a more difficult case is that F is strictly decreasing on $K\left(F\right)$. In this paper we discuss the decreasing case, giving existence of f of order $n<N\left(F\right)$ with $H\left(f\right)=n$ and of order $n\le N\left(F\right)+1$ with $H\left(f\right)=n-1$.