A sufficient condition for existence of iterative roots of PM functions without the characteristic endpoints condition

TL;DR

This paper addresses the characteristic endpoints problem for PM functions, providing a sufficient condition for the existence of continuous iterative roots of order 2 when the number of forts exceeds the order.

Contribution

It offers a new sufficient condition for iterative roots of PM functions without the characteristic endpoints condition, especially when the number of forts is greater than the order.

Findings

Established a sufficient condition for iterative roots of order 2

Solved the characteristic endpoints problem in a new case

Extended understanding of iterative roots beyond previous conditions

Abstract

For PM functions of height 1, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition. This raises an open problem about iterative roots without this condition, called characteristic endpoints problem. This problem is solved almost completely when the number of forts is equal to or less than the order. In this paper, we study the case that the number of forts is greater than the order and give a sufficient condition for existence of continuous iterative roots of order $2$ with height 2, answering the characteristic endpoints problem partially.