A classification of non-monotonicity height for piecewise monotone functions (I): increasing case

Although there are plentiful results on the dynamics of monotone mappings, the problem becomes difficult in the non-monotone case. For PM functions (i.e., for piecewise monotone functions), it is known that the non-monotonicity height is an important index to describe their complexity, and the dynamical properties for such functions are complicated when the non-monotonicity height is infinity. In this paper, we consider the non-monotonity height for a general PM function, and give a classification for the non-monotonicity height of those functions. Our results present a full description of their complexity under iteration.