Math 2.0

Dynamical systems and ergodic theory Topological dynamics Theorem

Sharkovskii's theorem

Builds on IntervalContinuous mapPeriodic pointMinimal period

Statement

Order the positive integers by the Sharkovskii ordering ▹\triangleright: 3▹5▹7▹⋯▹2⋅3▹2⋅5▹⋯▹22⋅3▹22⋅5▹⋯▹23▹22▹2▹1.3 \triangleright 5 \triangleright 7 \triangleright \cdots \triangleright 2 \cdot 3 \triangleright 2 \cdot 5 \triangleright \cdots \triangleright 2^2 \cdot 3 \triangleright 2^2 \cdot 5 \triangleright \cdots \triangleright 2^3 \triangleright 2^2 \triangleright 2 \triangleright 1. First come the odd numbers greater than 11 in increasing order, then 22 times them, then 222^2 times them, and so on. At the end come the powers of 22 in decreasing order. Let I⊆ℝI \subseteq \R be an interval and let f ⁣:I→If \colon I \to I be a continuous map.

  1. If ff has a periodic point of minimal period mm and m▹nm \triangleright n, then ff has a periodic point of minimal period nn.
  2. Conversely, for each integer m≥1m \ge 1 some continuous f ⁣:ℝ→ℝf \colon \R \to \R has a point of minimal period mm but none of minimal period kk when k▹mk \triangleright m.

Remarks

The proof of part 1 uses the intermediate value theorem in the form: if f(J)⊇Kf(J) \supseteq K for closed intervals J,KJ, K, then JJ contains a closed interval mapped onto KK. The points of a periodic orbit cut the interval into pieces, and a directed graph records which pieces cover which. Cycles in this graph give periodic points of the required periods. Example: since 33 is the first number in the ordering, a continuous map of an interval with a point of period 33 has points of every period. The hypothesis that the space is an interval is necessary: the rotation of the circle by 1/31/3 has every point of period 33 and no other periods. For a continuous map of a compact interval, the set of minimal periods is therefore of one of two kinds. It is either a tail {k:m⊵k}\set{k : m \trianglerighteq k} of the ordering for some mm, or the set of all powers of 22.

Going deeper

For a student

The theorem is purely one-dimensional: its proof uses the order of the real line through the intermediate value theorem, and it fails on the circle and in the plane. It refines the Li–Yorke result that period three implies chaos. In families like the logistic map, the passage from maps with only power-of-two periods to the others is where period doubling turns into chaos.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Continuous map, concept, no evidence levelContinuous mapconceptInterval, concept, no evidence levelIntervalconceptMinimal period, concept, no evidence levelMinimal periodconceptPeriodic point, concept, no evidence levelPeriodic pointconceptSharkovskii's theorem, statement, no evidence levelSharkovskii's theoremstatement
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  1. Continuous map concept
  2. Interval concept
  3. Minimal period concept
  4. Periodic point concept
  5. Sharkovskii's theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Sharkovskii%27s_theorem
About this page
TagYBCZ9VY7SX
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Kindstatement, theorem
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MSC37E15, 37E05
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AboutPeriodic point
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