A Remarkable Property of Real-Valued Functions on Intervals of the Real Line

Today the 17 October 2019 I discussed a very remarkable fixed point theorem discovered by the Ukrainian mathematician Oleksandr Micholayovych Sharkovsky.

We recall that a periodic point of period n\geq1 for a function f:X\to{X} is a point x_n such that f^n(x_n)=x_n. With this definition, a periodic point of period n is also periodic of period m for every m which is a multiple of n. If f^n(x_n)=x_n but f^k(x_n)\neq{x_n} for every k from 1 to n-1, we say that n is the least period of x_n.

Theorem 1. (Sharkovsky’s “little” theorem) Let I\subseteq\mathbb{R} be an interval and let f:I\to\mathbb{R} be a continuous function su. If f has a point of least period 3, then it has points of arbitrary least period; in particular, it has a fixed point.

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