Iterative roots of piecewise monotonic functions of nonmonotonicity height not less than 2

It is known that any piecewise monotonic function of nonmonotonicity height not less than 2 has no continuous iterative roots of order greater than the number of forts of the function. So the following problem arises naturally: does such a function have an iterative root of order n not greater than the number of forts? We consider the case that the number of forts is equal to n, in which there appear possibly only two types T-1 and T-2 of iterative roots, i.e., the roots strictly increasing on the interval stretched on all forts of the given function and the roots strictly decreasing on such an interval, respectively. We characterize all type T-1 roots of order n and give a necessary condition for a root of order n to be of type T-2. A full description of type T-2 is still an open question. (C) 2011 Elsevier Ltd. All rights reserved.