Sharkovskii's theorem

The set of Sarkovskii is a set of mathematics that makes an important statement about the possible periods in the iteration of a continuous function. A special case of the theorem is the statement that a continuous dynamical system on the real line has been a point of order 3 items for each order. This is often briefly formulated to period 3 implies chaos.

The set

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A continuous function. We say that x is a periodic point of order ( or period length ) m, if f m (x) = x ( where f m is the m -fold join of f with itself) and f k ( x ) ≠ x for all 0 < k < m. In the statement it comes to the possible orders of periodic points of f to its formulation we consider the so-called Sarkovskii - order of the natural numbers. It involves the total order

Thus, this sequence starts with the odd numbers in ascending order, followed by two times the odd numbers, the four times the odd numbers, etc., and ends with the powers of two in descending order.

The set of Sarkovskii now states that if f has a periodic point of length m and m ≤ n in the Sarkovskii - order that it then also (at least) is a periodic point of length n.

Conclusions and remarks

The theorem has several consequences. For one, if f has only finitely many periodic points, so they must all have a power of two as fine. If f has any periodic point, then f has also a fixed point. Furthermore, as soon as there is a point of order 3, so there are periodic points of every order. This statement is also called the set of Li and Yorke.

The set of Sarkovskii is optimal in the sense that one can construct m is a continuous function for every natural number such that there is any natural number that comes in the Sarkovskii order by m (including m), periodic points with this period length but no periodic points with a lower order. So there are, for example, functions that do not have periodic points of length 3, but probably to all other numbers ( period 5 does not imply chaos).

The set of Sarkovskii does not apply to dynamic systems on other topological spaces. For the rotation of the circular line 120 degrees (one-third rotation), each point is periodic with the length of 3, and no further period lengths appear.

History

This theorem was proved in 1964 by the Ukrainian mathematician Oleksandr Mikolaiovich Sarkovskii and remained a long time unnoticed. Some 10 years later, Li and Yorke proved without knowledge of the original result to the special case, that period implies chaos 3.