This paper is concerned with qualitative behaviors of iterates of functions. More specifically, given a function \(f:E\longrightarrow E\) where \(E\) is a nonempty set, the \(n\)-th iterate \(f^{n}\) is defined by \(f^{0}(x)=x,\) \(f^{n+1}(x)=f\left(f^{n}(x)\right)\) for all \(x\in E\). Let \(E\) be a real interval \(I\), and let \(f\) be a continuous map \(f:I\longrightarrow I\). A point \(a\in I\) is called a monotone point of \(f\) if \(f\) is strictly monotone in a neighborhood of \(a;\) otherwise it is called a fort of \(f\). Let \(S(f)\) be the set of all forts of \(f\) and let \(N(f)\) the cardinality of \(S(f)\).
There are many interesting questions that we can ask about the sequence \(\left\{ N(f^{n})\right\}_{n=0}^{\infty}\). It is obviously non-negative, but is it monotone, bounded, or tending to a limit?
In the case when \(f\) is a polynomial, based on the so-called “polynomial complete discrimination system”, this paper offers an algorithm to compute \(\left\{ N(f^{n})\right\}_{n=0}^{\infty}\). Then this algorithm is applied to the special quadratic function \(f(x)=g_{\mu}(x)=x^{2}+\mu\). It is shown that if \(\mu \geq 0\), then \(N\left(g_{\mu}^{n}\right) =N\left(g_{\mu}\right) =1\) for \(n\geq 2;\) and if \(\mu <0,\) then \(\lim_{n}N\left(g_{\mu}^{n}\right) =\infty\). Futhermore, for fixed \(n=2,3,\dots,7,\) it is shown that each \(N\left(g_{\mu}^{n}\right) ,\) as a function of \(\mu\), is constant over specific intervals of the form shown below:
\[ \begin{matrix} & N\left(g_{\mu}\right) & N\left(g_{\mu}^{2}\right) & N\left(g_{\mu}^{3}\right) & N\left(g_{\mu}^{4}\right) & N\left(g_{\mu}^{5}\right) & N\left(g_{\mu}^{6}\right) & N\left(g_{\mu}^{7}\right) \\ [ \mu_{2,0},+\infty ) & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ [ \mu_{3,1},\mu_{2,0}) & 1 & 3 & 5 & 7 & 9 & 11 & 13 \\ [ \mu_{5,1},\mu_{3,1}) & 1 & 3 & 7 & 13 & 21 & 31 & 43 \\ [ \mu_{7,1},\mu_{5,1}) & 1 & 3 & 7 & 13 & 23 & 37 & 57 \\ [ \mu_{6,1},\mu_{7,1}) & 1 & 3 & 7 & 13 & 23 & 37 & 59 \\ [ \mu_{4,1},\mu_{6,1}) & 1 & 3 & 7 & 13 & 23 & 39 & 65 \\ [ \mu_{7,2},\mu_{4,1}) & 1 & 3 & 7 & 15 & 29 & 53 & 93 \\ [ \mu_{6,2},\mu_{7,2}) & 1 & 3 & 7 & 15 & 29 & 53 & 95 \\ [ \mu_{7,3},\mu_{6,2}) & 1 & 3 & 7 & 15 & 29 & 55 & 101 \\ [ \mu_{5,2},\mu_{7,3}) & 1 & 3 & 7 & 15 & 29 & 55 & 103 \\ [ \mu_{7,4},\mu_{5,2}) & 1 & 3 & 7 & 15 & 31 & 61 & 117 \\ [ \mu_{6,3},\mu_{7,4}) & 1 & 3 & 7 & 15 & 31 & 61 & 119 \\ [ \mu_{7,5},\mu_{6,3}) & 1 & 3 & 7 & 15 & 31 & 63 & 125 \\ (-\infty ,\mu_{7,5}) & 1 & 3 & 7 & 15 & 31 & 63 & 127 \\ \end{matrix} \]
This table suggests several meaningful conjectures, one of which is that the numbers of changes of constancies of \(N(g_{\mu}^{n})\) match the Fibonacci numbers. Further investigations including symbolic computations of \(N(g_{\mu}^{n})\) are also of interest.