Sharkovskii theorem (period three implies all periods)

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Problem X.1.30 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

The set {xn=fn(x0)∣n∈N}\left\{ x_{n}=f^{n}\left(x_{0}\right) \mid n \in \mathbb {N}\right\} is called the trajectory of the point x0x_{0}. If the trajectory is finite, then there exists n∈Nn \in \mathbb {N} such that the points x0,x1,…,xn−1x_{0}, x_{1}, \ldots , x_{n-1} are pairwise distinct, while xn=x0,xn+1=x1,…x_{n}=x_{0}, x_{n+1}=x_{1}, \ldots; in this case x0x_{0} is called a periodic point, the number nn its period, and the tuple (x0,…,xn−1)\left(x_{0}, \ldots , x_{n-1}\right) an nn-cycle.

Prove that if a continuous map of a segment into itself has a 4-cycle, then it also has a 2-cycle.

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Problem X.1.31 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

The set {xn=fn(x0)∣n∈N}\left\{ x_{n}=f^{n}\left(x_{0}\right) \mid n \in \mathbb {N}\right\} is called the trajectory of the point x0x_{0}. If the trajectory is finite, then there exists n∈Nn \in \mathbb {N} such that the points x0,x1,…,xn−1x_{0}, x_{1}, \ldots , x_{n-1} are pairwise distinct, while xn=x0,xn+1=x1,…x_{n}=x_{0}, x_{n+1}=x_{1}, \ldots; in this case x0x_{0} is called a periodic point, the number nn its period, and the tuple (x0,…,xn−1)\left(x_{0}, \ldots , x_{n-1}\right) an nn-cycle.

Let f:Δ→Δf: \Delta \rightarrow \Delta be a continuous map of a segment into itself. Prove that the following statements are equivalent:

  1. there exists a point a∈Δa \in \Delta such that the points b=f(a),c=f2(a)b=f(a), c=f^{2}(a), d=f3(a)d=f^{3}(a) satisfy the inequalities d⩽a<b<cd \leqslant a<b<c or c<b<a⩽dc<b<a \leqslant d;

  2. in Δ\Delta there exists a point having period three;

  3. for any m∈Nm \in \mathbb {N}, in Δ\Delta there exists a point having period mm.

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Remark.
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The implication b) ⇒ c) and problem 1.30 are special cases of A.N. Sharkovsky's theorem [Sh]. Let f:Δ→Δf: \Delta \rightarrow \Delta be a continuous map (Δ=[a,b])(\Delta =\left[a, b\right]). Consider the natural numbers ordered as follows:

1,2,4,8,…,2n,…,…,2m⋅7,2m⋅5,2m⋅3,…,2⋅7,2⋅5,2⋅3,…,7,5,3. 1,2,4,8, \ldots , 2^{n}, \ldots , \ldots , 2^{m} \cdot 7,2^{m} \cdot 5,2^{m} \cdot 3, \ldots , 2 \cdot 7,2 \cdot 5,2 \cdot 3, \ldots , 7,5,3 .

If pp stands to the left of qq in this sequence and ff has a qq-cycle, then ff also has a pp-cycle.

Problem X.1.42 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

The set {xn=fn(x0)∣n∈N}\left\{ x_{n}=f^{n}\left(x_{0}\right) \mid n \in \mathbb {N}\right\} is called the trajectory of the point x0x_{0}. If the trajectory is finite, then there exists n∈Nn \in \mathbb {N} such that the points x0,x1,…,xn−1x_{0}, x_{1}, \ldots , x_{n-1} are pairwise distinct, while xn=x0,xn+1=x1,…x_{n}=x_{0}, x_{n+1}=x_{1}, \ldots; in this case x0x_{0} is called a periodic point, the number nn its period, and the tuple (x0,…,xn−1)\left(x_{0}, \ldots , x_{n-1}\right) an nn-cycle.

Let φ\varphi be a continuous map of the circle into itself having a fixed point and a 3-cycle.

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(a)

Prove that φ\varphi also has nn-cycles for all n>3n>3.

(b)

Must φ\varphi have a 2-cycle?

References

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.