17 October 2025
C. Hinsley
I’m publishing here some numerical results I’ve obtained tonight about the dynamics of one-dimensional maps that I found surprising. They're not groundbreaking or anything (you could have known them many decades ago easily) but I found them counterintuitive and I haven't seen anyone mention these facts before.
Let $f: I \to I$ be a continuous endomorphism of an interval having a single extremal point $c$ in the interior of $I$ such that $c$ is a maximal point:
$$ \argmax_{x \in I} f(x) = c. $$
We say that $f$ is unimodal because there is only one local extremal point on the interior of its domain. Suppose there exists $n \in \mathbb{Z}+$ such that $f^k(c) = c$ for $k = n$ and $f^k(c) \neq c$ for $k < n$. Then we say that $c$ is a periodic point of period $n$, or that $c$ belongs to an $n$-cycle $(c, c_1, c_2, \ldots, c{n-1})$ where $c_k = f^k(c)$ for each $k$. It turns out that, because $f$ is unimodal, there are only two possible orderings of the points in the cycle: (October 31st) This is wrong: You can have e.g. an $8$-cycle with $c_2 < c_6 < c < c_4 < c_3 < c_7 < c_5 < c_1$. But these kinds of discrepancies seem to only occur for renormalizable maps (special cases with $n = 8, 16, 32, \ldots$); I’ll verify at some point in the future. In any case, the two possibilities below are still interesting to consider, being generally satisfiable:
$$ c_2 < c_3 < \ldots < c_{n-1} < c < c_1 \quad \text{or} \quad c_2 < c_3 < \ldots < c_{n-2} < c < c_{n-1} < c_1. $$
We will refer to $n$-cycles as $n^-$-cycles if they satisfy the first inequality or $n^+$-cycles if they satisfy the second.
Remark 1. No $3^+$-cycle exists. That is, every $3$-cycle is a $3^-$-cycle.
Let $\ell_k$ denote the minimum number of subintervals $I_1, I_2, \ldots, I_{\ell_k} \subset I$ into which $I$ can be partitioned ($I = I_1 \cup \ldots \cup I_{\ell_k}$) such that $f^k$ is monotone on each $I_i$, $i = 1, 2, \ldots, \ell_k$. Then we define the topological entropy of $f$ **to be the quantity
$$ h_\mathrm{top}(f) = \lim_{k \to \infty} \frac{\ln\ell_k}{k}. $$
It can be shown (Misiurewicz & Szlenk 1980) that $h_\mathrm{top}(f)$ is zero if all periodic points are of period $2^m$ for some $m \in \mathbb{Z}{\geq 0}$; otherwise, $h\mathrm{top}(f) > 0$.
Remark 1. People commonly think of positive topological entropy as being synonymous with chaos-like dynamics, but in fact, if $f$ has periodic points of periods all powers of two and no others, its topological entropy is zero, but it still may behave unpredictably. Sylvie Ruette’s book talks about this (Sections 5.4 and 5.7) if you are interested in the details.
Remark 2. You will find it helpful to read about the Sharkovsky ordering before proceeding.
Let $\mathrm{Ш}(f) \in \mathbb{Z}_+$ denote the Sharkovsky type of the map $f$; i.e., the period of the Sharkovsky-maximal cycle in the domain of $f$.
It is known that two maps (unimodal or not) $f, g$ satisfy $h_\mathrm{top}(f) < h_\mathrm{top}(g)$ if $2^\infty \prec Ш(f) \prec Ш(g)$. But even though the topological entropy of $f$ may be determined purely by studying the forward itinerary of $c$ under $f$ (cf. Milnor-Thurston kneading theory), it is not true that the Sharkovsky-maximal cycle in the domain of $f$ attracts $c$! I will give a table of calculations below that make this clear.