Low-dimensional dynamics

Low-dimensional dynamics covers maps of the interval and circle, continued-fraction-type maps, flows in the plane and homeomorphisms of surfaces.

• Linearization of circle diffeomorphisms
• N-continued fraction expansions
• Natural duals of Möbius maps
• Poincaré–Bendixson theorem
• Simplicity conjecture

TheoremLi–Yorke theorem: period three implies all periods#✓fully proved

Let \(f \colon [0, 1] \to [0, 1]\) be a continuous map. If some point of \([0, 1]\) has minimal period \(3\) under \(f\), then for every positive integer \(n\) some point of \([0, 1]\) has minimal period \(n\) under \(f\).

theorem liYorke_period_three_implies_periods
    (f : Set.Icc (0 : ℝ) 1 → Set.Icc (0 : ℝ) 1)
    (hf : Continuous f)
    (h3 : ∃ x, Function.minimalPeriod f x = 3) :
    ∀ (n : ℕ), 0 < n → ∃ y, Function.minimalPeriod f y = n

Sources

• T.-Y. Li and J. A. Yorke, Period Three Implies Chaos, Amer. Math. Monthly 82 (1975), 985-992, DOI 10.2307/2318254.

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