Math 2.0

Dynamical systems and ergodic theory Low-dimensional dynamics Theorem

Milnor–Thurston kneading theory

Builds on Topological entropySemiconjugacy and factor mapFormal power seriesContinuous functionTent mapSharkovskii's theorem

Statement

Let f ⁣:[0,1]→[0,1]f \colon [0, 1] \to [0, 1] be a continuous unimodal map. So there is a turning point c∈(0,1)c \in (0, 1) such that ff is strictly increasing on [0,c][0, c] and strictly decreasing on [c,1][c, 1]. Assume that fn(c)≠cf^n(c) \neq c for all n≥1n \ge 1. For x∈[0,1]x \in [0, 1] let ε(x)=1\varepsilon(x) = 1 if x<cx < c, and ε(x)=−1\varepsilon(x) = -1 if x>cx > c. The kneading determinant of ff is the formal power series D(t)=1+∑n=1∞ε(f(c)) ε(f2(c))⋯ε(fn(c)) tn.D(t) = 1 + \sum_{n=1}^{\infty} \varepsilon(f(c)) \, \varepsilon(f^2(c)) \cdots \varepsilon(f^n(c)) \, t^n. Let ℓ(fn)\ell(f^n) be the number of maximal intervals on which fnf^n is monotone (the lap number of fnf^n). Then:

  1. The limit s=lim⁡n→∞ℓ(fn)1/ns = \lim_{n \to \infty} \ell(f^n)^{1/n} exists, 1≤s≤21 \le s \le 2, and the topological entropy of ff is h(f)=log⁡sh(f) = \log s.
  2. The power series D(t)D(t) converges for ∣t∣<1\abs{t} < 1 and has no zero in the disc ∣t∣<1/s\abs{t} < 1/s.
  3. If s>1s > 1, then D(1/s)=0D(1/s) = 0, so 1/s1/s is the smallest positive zero of DD.
  4. If s>1s > 1, then ff is semiconjugate to a piecewise linear unimodal map gg whose slopes are ss and −s-s. That is, there is a continuous nondecreasing surjection h ⁣:[0,1]→[0,1]h \colon [0, 1] \to [0, 1] with h∘f=g∘hh \circ f = g \circ h.

Remarks

The coefficients of D(t)D(t) are ±1\pm 1, which gives convergence for ∣t∣<1\abs{t} < 1. The sign sequence ε(f(c)),ε(f2(c)),…\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 T(x)=1−∣1−2x∣T(x) = 1 - \abs{1 - 2x} with c=1/2c = 1/2. Here T(c)=1>cT(c) = 1 > c and Tn(c)=0<cT^n(c) = 0 < c for n≥2n \ge 2, so the signs are −1,1,1,…-1, 1, 1, \dots and D(t)=1−t−t2−t3−⋯=1−2t1−t.D(t) = 1 - t - t^2 - t^3 - \cdots = \frac{1 - 2t}{1 - t}. Its only zero is t=1/2t = 1/2, and indeed ℓ(Tn)=2n\ell(T^n) = 2^n, so s=2s = 2 and h(T)=log⁡2h(T) = \log 2.

Going deeper

For a student

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↦rx(1−x)x \mapsto r x (1 - x) is a nondecreasing function of rr. It also explains the order in which periodic orbits appear in such families, which refines Sharkovskii's theorem.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Borel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptContinuous function, concept, no evidence levelContinuous functionconceptContinuous map, concept, no evidence levelContinuous mapconceptDiscrete-time dynamical system, concept, no evidence levelDiscrete-time dynamic…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectFormal power series, concept, no evidence levelFormal power seriesconceptInterval, concept, no evidence levelIntervalconceptInvariant measure, concept, no evidence levelInvariant measureconceptLyapunov exponent, concept, no evidence levelLyapunov exponentconceptMinimal period, concept, no evidence levelMinimal periodconceptOuter measure, concept, no evidence levelOuter measureconceptPeriodic point, concept, no evidence levelPeriodic pointconceptSemiconjugacy and factor map, concept, no evidence levelSemiconjugacy and fac…conceptTopological entropy, concept, no evidence levelTopological entropyconceptLebesgue measure, object, no evidence levelLebesgue measureobjectSharkovskii's theorem, statement, no evidence levelSharkovskii's theoremstatementTent map, example, no evidence levelTent mapexampleMilnor–Thurston kneading theory, statement, no evidence levelMilnor–Thurston knead…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel sigma-algebra concept
  2. Continuous function concept
  3. Continuous map concept
  4. Discrete-time dynamical system concept
  5. Euclidean space object
  6. Formal power series concept
  7. Interval concept
  8. Invariant measure concept
  9. Lyapunov exponent concept
  10. Minimal period concept
  11. Outer measure concept
  12. Periodic point concept
  13. Semiconjugacy and factor map concept
  14. Topological entropy concept
  15. Lebesgue measure object
  16. Sharkovskii's theorem statement
  17. Tent map example
  18. Milnor–Thurston kneading theory statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Milnor%E2%80%93Thurston_kneading_theory
About this page
Tag3N54R6HR05
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Kindstatement, theorem
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MSC37E05, 37B40, 37E15
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AboutTopological entropy
UsesTopological entropy, Semiconjugacy and factor map, Formal power series, Continuous function, Tent map, Sharkovskii's theorem
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