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The Sharkovsky Theorem: A Natural Direct Proof



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The Sharkovsky Theorem A Natural Direct Proof Keith Burns and Boris Hasselblatt Abstract We give a natural and direct proof of a famous result by Sharkovsky that gives a complete description of possible sets of periods for interval maps The new ingredient is the use of S tefan sequences 1 INTRODUCTION In this note f is a continuous function from an interval into itself The interval need not be closed or bounded although this is usually assumed in the literature The point of view of dynamical systems is to study iterations of f if f n denotes the n fold composition of f with itself then for a given point x one investigates the sequence x f x f 2 x f 3 x and so on This sequence is called the f orbit of x or just the orbit of x for short It is particularly interesting when this sequence repeats In this case we say that x is a periodic point and we refer to the number of distinct points in the orbit or cycle O f n x n 0 1 as the period of x 1 Equivalently the period of x is the smallest positive integer m such that f m x x A fixed point is a periodic point of period 1 that is a point x such that f x x A periodic point with period m is a fixed point of f m and of f 2m f 3m Thus if f n x x then the period of x is a factor of n If f has a periodic point of period m then m is called a period for or of f Given a continuous map of an interval one may ask what periods it can have The genius of Alexander Sharkovsky lay in realizing that there is a structure to the set of periods 1 1 The Sharkovsky Theorem The Sharkovsky Theorem involves the following ordering of the set N of positive integers which is now known as the Sharkovsky ordering 3 5 7 2 3 2 5 2 7 22 3 22 5 22 7 23 22 2 1 This is a total ordering we write l r or r l whenever l is to the left of r It is crucial that the Sharkovsky ordering has the following doubling property l r if and only if 2l 2r 1 This is because the odd numbers greater than 1 appear at the left end of the list the number 1 appears at the right end



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