Sharkovskii's theorem

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In mathematics, Sharkovskii's theorem, named after Oleksandr Mikolaiovich Sharkovsky, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.

The theorem[edit]

Suppose

f : \mathbb{R} \to \mathbb{R}

is a continuous function. We say that the number x is a periodic point of period m if f m(x) = x (where f m denotes the composition of m copies of f) and having least period m if furthermore f k(x) ≠ x for all 0 < k < m. We are interested in the possible periods of periodic points of f. Consider the following ordering of the positive integers:

\begin{array}{cccccccc}
3 & 5 & 7 & 9 & 11 & \ldots & (2n+1)\cdot2^{0} & \ldots\\
3\cdot2 & 5\cdot2 & 7\cdot2 & 9\cdot2 & 11\cdot2 & \ldots & (2n+1)\cdot2^{1} & \ldots\\
3\cdot2^{2} & 5\cdot2^{2} & 7\cdot2^{2} & 9\cdot2^{2} & 11\cdot2^{2} & \ldots & (2n+1)\cdot2^{2} & \ldots\\
3\cdot2^{3} & 5\cdot2^{3} & 7\cdot2^{3} & 9\cdot2^{3} & 11\cdot2^{3} & \ldots & (2n+1)\cdot2^{3} & \ldots\\
 & \vdots\\
\ldots & 2^{n} & \ldots & 2^{4} & 2^{3} & 2^{2} & 2 & 1\end{array}

We start, that is, with the odd numbers in increasing order, then 2 times the odds, 4 times the odds, 8 times the odds, etc., and at the end we put the powers of two in decreasing order. Every positive integer appears exactly once somewhere on this list. Note that this ordering is not a well-ordering, since the set \{2^k \ \mid\ k \in \mathbb{N}\} doesn't have a least element. Sharkovskii's theorem states that if f has a periodic point of least period m and m precedes n in the above ordering, then f has also a periodic point of least period n.

As a consequence, we see that if f has only finitely many periodic points, then they must all have periods which are powers of two. Furthermore, if there is a periodic point of period three, then there are periodic points of all other periods.

Sharkovskii's theorem does not state that there are stable cycles of those periods, just that there are cycles of those periods. For systems such as the logistic map, the bifurcation diagram shows a range of parameter values for which apparently the only cycle has period 3. In fact, there must be cycles of all periods there, but they are not stable and therefore not visible on the computer generated picture.

Interestingly, the above "Sharkovskii ordering" of the positive integers also occurs in a slightly different context in connection with the logistic map: the stable cycles appear in this order in the bifurcation diagram, starting with 1 and ending with 3, as the parameter is increased. (Here we ignore a stable cycle if a stable cycle of the same order has occurred earlier.)

The assumption of continuity is important, as the discontinuous function f : x \mapsto (1-x)^{-1}, for which every value has period 3, would otherwise be a counterexample.

Generalizations[edit]

Sharkovskii's theorem does not immediately apply to dynamical systems on other topological spaces. It is easy to find a circle map with periodic points of period 3 only: take a rotation by 120 degrees, for example. But some generalizations are possible, typically involving the mapping class group of the space minus a periodic orbit.

References[edit]

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