A Note on Characteristic Endpoints Question for Decreasing Iterative Roots on Characteristic Interval

Abstract

For a piecewise monotone function F of height 1, an open question was raised: Does F have an iterative root f of order n ≤ N(F) + 1 if the ‘characteristic endpoints condition’ is not satisfied? This question was answered partly in the case that F is strictly increasing on its characteristic interval K(F) but f is strictly decreasing on K(F). In this paper we discuss the question for F increasing on K(F) in some remaining cases, giving the necessary and sufficient conditions for the existence of continuous iterative roots f decreasing on K(F) of order n = N(F) > 2 with H(f) = n − 1.