---
tag: 3N54R6HR05
kind: statement
role: theorem
name: Milnor–Thurston kneading theory
msc: ["37E05", "37B40", "37E15"]
uses: [BDZYBXBFDE, AEXJNRBASV, 033WGN00BF, 0X3WGX95FE, D0A22MEB1F, YBCZ9VY7SX]
about: [BDZYBXBFDE]
sources:
  - url: https://en.wikipedia.org/wiki/Milnor%E2%80%93Thurston_kneading_theory
same_as:
  wikidata: Q17103861
  wikipedia: Milnor–Thurston kneading theory
---

## Statement

Let $f \colon [0, 1] \to [0, 1]$ be a [[continuous|0X3WGX95FE]] unimodal map.
So there is a turning point $c \in (0, 1)$ such that $f$ is strictly increasing on $[0, c]$ and strictly decreasing on $[c, 1]$.
Assume that $f^n(c) \neq c$ for all $n \ge 1$.
For $x \in [0, 1]$ let $\varepsilon(x) = 1$ if $x < c$, and $\varepsilon(x) = -1$ if $x > c$.
The *kneading determinant* of $f$ is the [[formal power series|033WGN00BF]]
$$D(t) = 1 + \sum_{n=1}^{\infty} \varepsilon(f(c)) \, \varepsilon(f^2(c)) \cdots \varepsilon(f^n(c)) \, t^n.$$
Let $\ell(f^n)$ be the number of maximal intervals on which $f^n$ is monotone (the lap number of $f^n$).
Then:

1. The limit $s = \lim_{n \to \infty} \ell(f^n)^{1/n}$ exists, $1 \le s \le 2$, and the [[topological entropy|BDZYBXBFDE]] of $f$ is $h(f) = \log s$.
2. The power series $D(t)$ converges for $\abs{t} < 1$ and has no zero in the disc $\abs{t} < 1/s$.
3. If $s > 1$, then $D(1/s) = 0$, so $1/s$ is the smallest positive zero of $D$.
4. If $s > 1$, then $f$ is [[semiconjugate|AEXJNRBASV]] to a piecewise linear unimodal map $g$ whose slopes are $s$ and $-s$. That is, there is a continuous nondecreasing surjection $h \colon [0, 1] \to [0, 1]$ with $h \circ f = g \circ h$.

## Remarks

The coefficients of $D(t)$ are $\pm 1$, which gives convergence for $\abs{t} < 1$.
The sign sequence $\varepsilon(f(c)), \varepsilon(f^2(c)), \dots$, the kneading sequence, records on which side of the turning point the orbit of the critical value falls.
Milnor and Thurston show that it determines the combinatorics of all orbits: the lap numbers, the numbers of periodic points, and the entropy can all be read off from it.

Example: the full [[tent map|D0A22MEB1F]] $T(x) = 1 - \abs{1 - 2x}$ with $c = 1/2$.
Here $T(c) = 1 > c$ and $T^n(c) = 0 < c$ for $n \ge 2$, so the signs are $-1, 1, 1, \dots$ and
$$D(t) = 1 - t - t^2 - t^3 - \cdots = \frac{1 - 2t}{1 - t}.$$
Its only zero is $t = 1/2$, and indeed $\ell(T^n) = 2^n$, so $s = 2$ and $h(T) = \log 2$.

## Explanations

### L1 human

A unimodal map folds an interval once, like kneading dough.
Everything about its orbits is encoded in one list of signs: where the image of the fold point lands, left or right of the fold, after each step.
Milnor and Thurston turned this list into a power series and showed that its first zero gives the entropy, the exponential rate at which the map creates complexity.

### L2 human

The theorem reduces the topological dynamics of unimodal maps, such as the logistic family, to symbolic data.
Part 4 says that, up to collapsing some intervals, every unimodal map with positive entropy is a tent-like map of constant slope, so the entropy is visible as a slope.
Kneading theory was a main tool in the proof that topological entropy of the logistic family $x \mapsto r x (1 - x)$ is a nondecreasing function of $r$.
It also explains the order in which periodic orbits appear in such families, which refines [[Sharkovskii's theorem|YBCZ9VY7SX]].