Non-monotonicity height for cubic polynomials

Non-monotonicity height describes complexity of a non-monotonic continuous function under iteration by presenting the change of the number of non-monotonic points of the function under iteration. It was proved that the height of any quadratic polynomial, a special continuous function, is either 1 or infinity, i.e., the number of non-monotonic points is either constant 1 or increases to infinity as the number of iteration increases. For the latter case, they calculated the parameters at which the number of non-monotonic points varies and conjectures that the number of non-monotonic points increases in the law 2 ⁿ-1 for parameter μ＜ μ _7.5. and that the non-monotonic height for cubic polynomials is investigated and the height is proved either 0 or infinity. Additionally, a negative answer is given to the predecessors' conjecture for quadratic polynomials.