The question on characteristic endpoints for iterative roots of PM functions

Iterative roots of mappings are of special interests because it defines fractional iteration, displays middle procedure of evolution and proposes a weak version of the embedding flow problem. For PM functions of height 1, the class of 1-dimensional mappings having the simplest nonmonotonicity, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition and the condition was proved to be necessary for those orders greater than the number of forts plus 1. This suggests an open question about iterative roots without that condition, called the question on characteristic endpoints. In this paper, the question is answered completely in the case that the number of forts is equal to the order. Although results of nonexistence are also obtained for the case that the number of forts is greater than the order, a full description is still open.